Lecture Outlines
We will cover a wide variety of materials during lecture and discussion sections, so your constant attendance is important. To help you in organizing your study materials, the list below gives an overview of the basic concepts covered during a given lecture period.
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After covering some basic logistics related to the class, we focused our mathematical attention on logical statements. For a handful of sentences we determined whether they qualified as logical statements. We discussed connectives and modifiers for logical statements. These provide a way to take old statements and create new statements out of them. Specifically, we introduced the connectives $\vee, \wedge, \Rightarrow$ and $\Leftrightarrow$, as well as the negation modifier $\neg$. We saw how these interact with each other in both DeMorgan's laws and the contrapositive law.
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At the start of today's class we introduced the notion of a quantified statement. Such a statement begins with a list of quantified variables, and finishes with a predicate. The two types of quantifiers are existential ($\exists$) and universal ($\forall$). We saw that the order of quantified variables in a statement matters (because, for instance, $\forall x \in \mathbb{R} \exists y \in \mathbb{R}: x+y=0$ is different from the statement $\exists y \in \mathbb{R} \forall x \in \mathbb{R}: x+y = 0$), and we also discussed (and implemented) a process for negating quantified statements.
Next we introduced the notion of a field. A field $\mathbb{F}$ is a set equipped with two "binary operations" $+$ and $\cdot$ that satisfy a whole list of axioms (i.e., basic properties). These captures many of the familiar rules of arithmetic you'll have seen previously in your mathematical life, like commutativity of addition and the distributive property. We stated that some familiar numbers systems --- $\mathbb{R}, \mathbb{C}, \mathbb{Q}$ --- are fields, whereas others are not. For instance, we saw that $\mathbb{Z}$ is not a field because not all integers have multiplicative inverses which are again integers. We also saw that the set of irrational numbers is not a field; it fails even more spectacularly, since addition and multiplication aren't binary operations on the set of irrationals.
At the very end of class we introduced the notion of a system of linear equations over a given field. It is solving systems of equations that will be our entryway into the theory of linear algebra.
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Today we spent some time developing some notation to more conveniently write out systems of equations. With this notation in hand, we were then ready to ask: when does a given system have a solution? if it does have a solution, is it unique? can we find an algorithm to describe all solutions to a given system? The first two will require some more work on our part, but we tackled the third today be considering the notion of elementary row operations. These are transformations one can perform on a system of equations which keep the solution set identical. The idea is to use these elementary row operations to replace a given (complicated) system with another (simpler) system whose solutions are "easy" to determine. In practice, we said the kind of "easy" matrix we're aiming for is one in reduced row echelon form. We carried out one specific example to see how this played out in detail.
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We defined the notion of row equivalence between matrices today. Our definition of row equivalence implies a certain "directionality" (namely, to say that $A$ is row equivalent to $B$ means we can transform $A$ into $B$; it doesn't say anything about transforming $B$ into $A$). We saw, however, that row equivalence is an equivalence relation: it is reflexive, symmetric and transitive.
We finished class by reminding ourselves of the big question for computing solutions to linear systems: for a given system $AX=B$, how do we know if we have solutions? We saw last time this can be answered by performing row operations so that $[A|B]$ is transformed into a "simpler" system; i.e., one in reduced row echelon form. We finished class by stating (and sketching a proof of) a theorem which tells us how to determine the number of solutions to the system that corresponds to $[A|B]$ by looking at pivots in the row-equivalent system $[E|H]$, where $E$ is a matrix in reduced row echelon form. We stated that (1) the system $AX=B$ has a solution if each row of $E$ has a pivot (2) the system $AX=B$ has no solutions if and only if there is a row of $[E|H]$ whose left-most nonzero entry is in the augmented column, and (3) if $AX=B$ has a solution, then the solution is unique if and only if column of $E$ has a pivot.
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Class started today with some computations for solving two specific systems of equations. These two systems had the same coefficient part, so our idea was to simplify the process of solving these systems by creating a doubly augmented matrix $[A|B_1 B_2]$. After row reducing, we could ignore the last column to find solutions to $AX = B_1$, and we could ignore the second-to-last column to compute solutions to $AX = B_2$. We stated that the procedures we've outlined in the past few classes are enough to solve any system of equations; if linear algebra were only about solving linear systems, we'd then be done. Instead, though, we'll spend the rest of the semester trying to capture deeper meaning from these linear systems. As a hint of what's to come, we considered the system $$\left\{\begin{array}{rl}x_1+2x_2&=4\\3x_3+5x_4&=2\end{array}\right\}.$$ Instead of solving the system using row reduction, we instead had the following weird idea: let $y_1 = x_1 + 2x_2$, and let $y_2 = 3x_3+5x_4$. (Note: if our original system is true, this means that $y_1 = 4$ and $y_2 = 2$.) Now we considered a new system $$\left\{\begin{array}{rl}-5y_1+2y_2&=-16\\3y_1-y_2&=10\end{array}\right\}.$$ (In this case, the values on the right hand side appeared because we plugged the values $y_1 = 4$ and $y_2 = 2$ from our old system into the left-side of our new system.) We then plugged our original definitions for $y_1$ and $y_2$ into this new system: $$\left\{\begin{array}{rl}-5(x_1+2x_2) + 2 (3x_1+5x_2) &=-16\\3(x_1+2x_2)-(3x_1+5x_2) &=10\end{array}\right\}.$$ Though this system seems pretty horrific, it actually simplifies into a very easy system, namely $$\left\{\begin{array}{rl}x_1&=-16\\x_2&=10\end{array}\right\}!$$ So this gives us a really clever way to solve our original system.
This process of "plugging one system of equations into another" gives rise to the operation of matrix multiplication. We defined row/column multiplication, and then used it to give a definition for a product of two matrices $A$ and $B$; the product $AB$ is defined when the number of columns of $A$ equals the number of rows of $B$. We computed a few example and saw that matrix multiplication has some strange properties, like: not all matrices commute; the product of nonzero matrices can be nonzero.
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In the first half of class today we thought about a number of properties related to matrix arithmetic; i.e., properties which hold under matrix multiplication, addition and scaling. One of the big results we had told us that when $AB$ is defined, then $\text{Col}_k(AB) = A\text{Col}_k(B)$, and a similar result for rows of $AB$. We also saw how to interpret $A\text{Col}_k(B)$ in terms of "linear combinations" of the columns of $A$. With these arithmetic results in hand, we returned to an old example to motivate the definition of inverse. We saw that if $A \in \mathbb{F}_{n \times n}$, then we say that a matrix $B \in \mathbb{F}_{n \times n}$ is an inverse for $A$ if $$AB = BA = I_n.$$ We proved that inverses are unique. We also saw that if $A$ is a matrix that has an inverse, then that inverse can be used to prove that $AX = B$ has a unique solution, and that unique solution must be $X = A^{-1}B$.
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After the quiz, we introduced a little terminology around matrix inverses; we defined nonsingular, for instance, as well as the collection $GL_\mathbb{F}(n)$. We stated a result that said that nonsingular matrices are closed under multiplication (which you'll check on your homework), and we thought a little bit about how this is not true when it comes to matrix addition. We then asked the question: how can we determine if a matrix is invertible? and if so, how do we compute its inverse. We spent the balance of the class working towards proving the following result: if $A$ is nonsingular, and if one reduces the augmented matrix $[A|I_n]$ to reduced row echelon form, then one is left with the augmented matrix $[I_n|P]$; our claim is that $P = A^{-1}$. In class we argued that $AP = I_n$.
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We picked up our discussion from last time by working to prove that if $[A|I] \sim_R [I|P]$, then we have $PA=I$. This would complete our proof that the matrix $P$ is indeed $A^{-1}$. To do this, we first defined the notion of elementary matrices. We argued that if $\rho$ is an elementary row operation, and if $\rho$ carries a matrix $A$ to a matrix $B$, then we must have $E_\rho A = B$. Using this we were able to
provecomplete part of a proof of the fact that elementary matrices are invertible, and indeed that $(E_\rho)^{-1} = E_{\rho^{-1}}$. With this in hand, we were able to prove the desired equality --- $PA = I$ --- by thinking about how the row operations which carry $[A|I]$ to $[I|P]$ can be encoded in terms of some matrix products. -
Today found us collecting together a few results about nonsingular matrices that we've listed out over the past many lectures. In particular, we wrote down a theorem which gives us many different properties which are equivalent to nonsingularity; these are phrased in terms of systems of equations, row reduction, and more. It is an EXCELLENT exercise to try to "connect the dots" in this proof, since it's a good way to keep track of how these various ideas interact with each other.
We then defined the transpose of a matrix, together with some relevant arithmetic properties. To be honest: we're not at a point yet in the course where we'll do much with transposes, but they will become useful to us later on.
We finished class by listing the axioms of a vector space. Importantly, remember that a vector space consists of a set $V$ (which we think of as the set of "vectors" in the vector space), a binary operation which we think of as "vector addition," a field $\mathbb{F}$ and a scaling operation $\cdot:\mathbb{F}\times V \to V$. These operations then must satisfy a number of axioms, each of which tells us some basic property of the operations.
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After the quiz we analyzed some properties that hold true for all vector spaces. We showed that the additive identity and additive inverses are unique, and we gave an efficient way to test is a given vector is $0_V$ (the "walks like a duck" theorem). We then argued that $c \cdot 0_V$ is always $0_V$ (regardless of the value of $c \in \mathbb{F}$ chosen); that $0\cdot v = 0_V$ regardless of the choice of $v \in V$; and that $-1_{\mathbb{F}} \cdot v = -v$.
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Today we provided a method for quickly checking whether a given nonempty subset of a vector space is again a vector space. This was the subspace test. We started by proving this result, which effectively meant that we front-loaded the work of verifying vector space axioms when a particular condition is met. We then used this condition to determine whether a few subsets were subspaces. For instance, we proved that $\{\mathbf{x} \in \mathbb{R}^4: x_1+x_2+x_3+x_4=0\}$ is a subspace of $\mathbb{R}^4$. We also used this to argue that if $V$ is any vector space, then $\{0_V\}$ is a vector space.
We then moved on to define the notions of linear combination and span. We saw that $0_V$ is in the span of any collection of vectors, and indeed that the span of any collection of vectors is a subspace of the ambient space the vectors are chosen from.
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We started class today by reviewing some of the concepts related to span from last class period. In particular, we did a specific example where we asked if a given matrix was in the span of some other specific matrices. In this particular problem, we saw that we could translate this question into a corresponding system of linear equations which we could then solve using our usual process. After this we commented that not all vector spaces can be realized as the span of a finite collection of vectors. To see this, we stated that the set of polynomials with coefficients in $\mathbb{F}$ form a vector space, and that any finite set of such vectors can't span the space because any linear combination of such polynomials has bounded degree.
We then started to analyze some subspaces that were closely related to matrices. We defined null space, row space and column space, and we gave an alternative way to describe these latter two sets in terms of matrix/vector products.
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Today we delved more deeply into the properties of null space, row space and column space, focusing in particular on how these sets behave under matrix multiplication. For instance, we saw that $NS(A) \subseteq NS(MA)$, and that equality occurs if and only if $NS(M) = \{0\}$. We were able to use this (and other) results to argue that null space and row space are preserved by elementary row operations.
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Today's class focused on analyzing the notions of linear dependence and independence. We gave several examples of linearly dependent sets, as well as examples of independent sets.
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We started class by discussing some properties related to linear independence and dependence. For instance, we said that if we take a set of linearly independent vectors and remove some vectors from that set, we're left with a linearly independent collection. We discussed the notion of bases, and thought carefully about some properties that bases enjoy. For instance, if $\{w_1,\cdots,w_n\}$ is a basis for a space $W$, then we know for each $w \in W$ there is a unique way to express $w$ as a linear combination of the basis $\{w_1,\cdots,w_n\}$. At the end of class we stated an important theorem: if $\{w_1,\cdots,w_n\}$ is a basis for a space $W$, then any collection of more than $n$ elements from $W$ must be dependent. While we didn't give a proof, we were able to use this to argue that if a space $W$ admits a finite basis, then any two bases for that space must have the same number of elements. We therefore defined the dimension of $W$ --- written $\dim(W)$ --- to be the number of vectors in a basis for $W$.
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At the start of class today we verified the key lemma necessary to ensure that the notion of dimension was sensible: that if $V$ has a basis with $n$ elements, and $\{v_1,\cdots,v_m\}$ is some collection of vectors with more than $m$ elements, then $\{v_1,\cdots,v_m\}$ is linearly dependent. We discussed some "standard bases" for some familiar vector spaces, and as a consequence were able to compute some relatively simple dimensions: $\dim(\mathbb{F}_{r \times c}) = rc$, for instance. We also gave a method for computing a basis for the row space of a given matrix.
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Building on our algorithm for computing a basis for the row space of a given matrix, in class today we started with providing algorithms for computing bases for null space and column space as well. These again depend on the reduced row echelon form of the matrix in question, though in a more subtle way than the basis for the row space. We were able to argue that $\dim(NS(A)) + \dim(CS(A))$ always equals the number of columns of $A$: a result known as the rank-nullity theorem. We defined the rank of a matrix to be the dimension of its column space; we were able to prove that this is also the same as the dimension of the row space of $A$ (and, indeed, the row and column spaces of $A^T$ as well!).
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Now that we have methods for computing bases for some familiar spaces, we began thinking about how we might measure when two vector spaces are "the same." To make this notion precise, we defined vector space isomorphism; an isomorphism of vector spaces is a function $f:V \to W$ which preserves both the addition and scaling operations of the relevant vector spaces, but which also provides a bijection between $V$ and $W$. That is to say, $f$ is both an injective and a surjective function. We gave examples of injective and surjective functions. We then went on to define a function from a vector space to a set of column vectors. This function depends on a choice of basis $\mathbb{B}$ for the vector space $V$, and we called it the coordinate function for $\mathbb{B}$. Specifically, we have $\text{Crd}_{\mathbb{B}}:V \to \mathbb{F}_{\dim(V)\times 1}$ is the function which sends any given $v \in V$ to the vector which records the coefficients necessary to write $v$ as a linear combination in $\mathbb{B}$. We argued that this function is legitimately a function (i.e., each input goes to a unique output), and furthermore we showed that it preserves the vector space operation of addition. We commented that the proof that it preserves scaling is analogous. Hence if we want to show that a coordinate function is an isomorphism, we only have to show that it is a bijection. We left that as a goal for next class period.
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At the start of class today we finished our proof that for any basis $\mathbb{B}$ on a vector space $V$, we have that the coordinate function $\text{Crd}_\mathbb{B}:V \to \mathbb{F}_{\dim(V) \times 1}$ is an isomorphism. In particular, we focused on proving that $\text{Crd}_\mathbb{B}$ is both injective and surjective. Along the way we saw templates for writing injectivity and surjectivity proofs. Ultimately we'd like to be able to argue that any two vector spaces over $\mathbb{F}$ of the same dimension are isomorphic to each other, but this requires us to first develop some machinery related to functions. Hence we spent the balance of our class thinking about some properties of functions. We defined the notion of function composition, and argued that the composition of injective functions is injective, and that the composition of surjective functions is surjective. (As an easy consequence, the composition of bijections is again a bijection.) We then thought about how injectivity and surjectivity relate to the notion of left- and right-inverses.
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The big concept we introduced today was the idea that ``isomorphic to" is an equivalence relation. Before we dove into the proof, we started by showing that this result tells us that dimension characterizes isomorphism: if $V$ and $W$ are two vector spaces over a field $\mathbb{F}$, then $V \simeq W$ if and only if $\dim(V) = \dim(W)$. The proof of this result in one direction --- that isomorphic spaces have the same dimension --- required us to faithfully translate properties of independence, injectivity, spanning, and surjectivity to argue that if $\{v_1,\cdots,v_n\}$ is a basis for $V$, and if $f:V \to W$ is an isomorphism, then $\{f(v_1),\cdots,f(v_n)\}$ is a basis for $W$. The proof in the other direction simply required us to use the fact that ``isomorphic to" is an equivalence relation. We finished class by defining some terminology that will be necessary check the symmetry of "isomorphic to," and hence we defined the notion of left and right inverses.
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Our goal is to prove that ``isomorphic to" is an equivalence relation, and at the start of class the last piece of the puzzle was to develop the right perspective to prove symmetry. In order to do this, we thought more carefully about inverse functions. We were able to argue that only bijections can be invertible, and that for a given bijection there can be at most one inverse. Indeed, we were able to determine precisely what that inverse function should be. In the case that $f:V \to W$ is an isomorphism, we were able to prove that the corresponding inverse function $f^{-1}:W \to V$ is also an isomorphism. We then proved the other properties necessary to check that "isomorphic to" is an equivalence relation.
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In class today we asked the question: can we find an easy way to convert coordinate representations between two different bases. A bit more specifically: can we find some function which sends $\text{Crd}_\mathbb{B}(v)$ to $\text{Crd}_\mathbb{D}(v)$ for all vectors $v$? We answered this by first asking if we could hope to find such a function via matrix multiplication, and from there attempted to decide properties a matrix $P$ would need to satisfy in order for $$P\text{Crd}_\mathbb{B}(v) = \text{Crd}_{\mathbb{D}}(v) \quad \text{ for all }v \in V.$$ This led us to conjecture that $$P = \left[\begin{array}{c|c|c}\ & & \ \\\text{Crd}_\mathbb{D}(b_1)&\cdots & \text{Crd}_\mathbb{D}(b_n)\\\ & &\ \end{array}\right],$$ where $\mathbb{B} = \{b_1,\cdots,b_n\}$. We then verified that this matrix really does perform the desired action. We went on to ask if we could easily find a matrix $Q$ that sent $\mathbb{D}$ coordinates to $\mathbb{B}$ coordinates, and we argued that $Q = P^{-1}$.
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Our next course topic is linear transformations. A linear transformation is simply a function $T:V\to W$ which preserves both addition and scaling. So, for example, all isomorphisms are linear transformations; they happen to be linear transformations that are bijections. We denoted the set of all linear transformations from $V$ to $W$ by $\mathcal{L}(V,W)$. We gave a lot of examples of linear transformations. We proved some theoretical properties about linear transformation, like any $T \in \mathcal{L}(V,W)$ must satisfy $T(0_V) = 0_W$, and $T(-v) = -T(v)$ for all $v \in V$. We then stated a theorem that told us that linear transformations are determined by their action on a basis of $V$, though we didn't get to prove this result.
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Much of the class was devoted to proving the result from last time that any linear transformation is determined by its action on a basis. In the time that was left in class, we considered how linear transformations behave relative to subspaces. We defined the notion of image and inverse image for a generic function, and saw that images of subspaces under linear transformations are again subspaces, and likewise inverse images of subspaces in the codomain under linear transformations are subspaces. We focused on two specific examples: the "full" image of $T$ (i.e., $T(V)$) and the kernel of $T$ (i.e., $T^{-1}(\{0_W\})$).
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We started class by giving a partial proof of our claim from last lecture: that if $T \in \mathcal{L}(V,W)$, then the image of subspaces from $V$ under $T$ are subspaces in $W$, and likewise the inverse image of subspaces from $W$ under $T$ are subspaces of $W$. We proved the "image" portion of this theorem. We then went on to define addition and scaling operations on $\mathcal{L}(V,W)$. For instance, if $T_1,T_2 \in \mathcal{L}(V,W)$, then we define $T_1+T_2:V \to W$ by $$(T_1+T_2)(v) = T_1(v) \oplus T_2(v) \quad \quad \text{ for all }v \in V.$$ We argued that the result of addition of scaling functions from $\mathcal{L}(V,W)$ produces linear transformations, and hence we realized that $\mathcal{L}(V,W)$ (under the defined operations of function addition and scaling) form a vector space!
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In today's class we asked: if $\mathcal{L}(V,W)$ is a vector space, then can we find any vector space to which is is naturally isomorphic? We were able to answer this quite nicely in the case of $\mathcal{L}(\mathbb{F}_{c \times 1},\mathbb{F}_{r \times 1})$, since we produced an explicit isomorphism $\psi:\mathbb{F}_{r \times c} \to \mathcal{L}(\mathbb{F}_{c \times 1},\mathbb{F}_{r \times 1})$ which assigns to $A \in \mathbb{F}_{r \times c}$ the function $f_A$ which multiplies on the left by $A$. In particular, this showed us that $$\dim\left(\mathcal{L}(\mathbb{F}_{c \times 1},\mathbb{F}_{r \times 1}\right) = rc.$$
Towards the end of class we worked to make this result applicable to all vector spaces, as follows. If $\mathbb{B} = \{v_1,\cdots,v_n\}$ is a basis for $V$ and $\mathbb{D} = \{w_1,\cdots,w_m\}$ is a basis for $W$, then we attempted to find some matrix to which we could associate a given linear transformation $T \in \mathcal{L}(V,W)$. One natural way to do this is to translate vectors in $V$ and $W$ into column vectors via coordinate isomorphisms. Form this perspective, we hope to find a matrix $\psi(T)$ so that $$\psi(T) \text{Crd}_\mathbb{B}(v) = \text{Crd}_\mathbb{D}(T(v)) \quad \quad \text{ for all }v \in V.$$ With this as inspiration, we defined a function $Mtx_{\mathbb{D},\mathbb{B}}:\mathcal{L}(V,W) \to \mathbb{F}_{\dim(W) \times \dim(V)}$ by setting $$Mtx_{\mathbb{D},\mathbb{B}}(T) = \left[\begin{array}{c|c|c}\ & \ &\ \\ \text{Crd}_\mathbb{D}(T(v_1)) & \cdots & \text{Crd}_\mathbb{D}(T(v_n))\\\ &\ &\ \end{array}\right].$$ We performed a few calculations to see how this played out in the context of the derivative operator on $\mathbb{F}_n[x]$.
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With last class as our motivation, we spent today working towards a proof that $\mathcal{L}(V,W) \simeq \mathbb{F}_{\dim(W) \times \dim(V)}$. The essential ingredient was in arguing that $$\text{Mtx}_{\mathbb{D},\mathbb{B}}(T)\text{Crd}_\mathbb{B}(v) = \text{Crd}_\mathbb{D}(T(v)).$$ This was the key technical lemma for proving that $\text{Mtx}_{\mathbb{D},\mathbb{B}}:\mathcal{L}(V,W) \to \mathbb{F}_{\dim(W)\times \dim(V)}$ satisfies the necessary properties to be an isomorphism: that it is injective, surjective, and preserves the appropriate operations. As a quick corollary we observed that this meant that $\dim(\mathcal{L}(V,W)) = \dim(v)\dim(W)$, an observation that we made earlier but weren't then able to prove.
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Once we untangled how matrix representations for a linear transformation act on coordinate vectors, the next question was to ask: how does matrix representation encode some of the "bonus" structure that $\mathbb{F}_{\dim(W)\times\dim(V)}$ and $\mathcal{L}(V,W)$ have. In particular, linear transformations can be composed, and matrices can be multiplied. We saw that these operations are indeed related under matrix representation: if $\mathbb{B}$, $\mathbb{D}$ and $\mathbb{E}$ are bases for $V,W$ and $U$, respectively, then for linear transformations $T \mathcal{L}(V,W)$ and $S \in \mathcal{L}(W,U)$ we have $$\text{Mtx}_{\mathbb{E},\mathbb{D}}(S)\text{Mtx}_{\mathbb{D},\mathbb{B}}(T) = \text{Mtx}_{\mathbb{E},\mathbb{B}}(S \circ T).$$ We got two nice corollaries from this. The first is that when $T$ is an isomorphism, we have $$\left(\text{Mtx}_{\mathbb{D},\mathbb{B}}(T)\right)^{-1} = \text{Mtx}_{\mathbb{B},\mathbb{D}}(T^{-1}).$$ The second is that we can deduce a relationship between matrix representations for $T$ in different bases; this gave us the "change of basis" formula $$\text{Mtx}_{\hat{\mathbb{D}},\hat{\mathbb{B}}}(T) = \text{Mtx}_{\hat{\mathbb{D}},\mathbb{D}}(\text{id}_W)\text{Mtx}_{\mathbb{D},\mathbb{B}}(T)\text{Mtx}_{\mathbb{B},\hat{\mathbb{B}}}(\text{id}_V).$$
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We started class by working through an example: how would we represent the action of reflection across the line spanned by $[3,5]^T$ in $\mathbb{R}^2$ (as a matrix in standard coordinates)? We saw that although this question was quite hard, it was much easier to represent reflection across the line spanned by $[1,0]^T$. With this as inspiration, we saw that we could find a basis $\mathbb{B}$ of $\mathbb{R}^2$ that was better suited for reflection across our original line, and in that basis the matrix representation for reflection was quite simple. Using our change of basis technology, this allowed us to express this reflection in standard coordinates (as a product of matrices). This motivated the definition of similarity of matrices, and the philosophical interpretation that matrices are similar if they represent the same linear transformation in different bases.
This also prompted a discussion of when certain transformations have "nice" bases for representing their action. We defined the notion of eigenvectors (or characteristic vectors) as well as eigenvalues.
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We spent the class today thinking about eigen-stuff more. We proved that a nonzero eigenvector can have at most one eigenvalue, and we defined the notion of an eigenspace $E_\lambda$ associated to $\lambda \in \mathbb{F}$. We proved that $E_\lambda$ is a subspace. We further observed that $E_0$ is simply $\ker(T)$. We discussed the notion of diagonalizability for matrices and linear transformations, and argued that a linear transformation is diagonalizable if and only if it admits an eigenbasis.
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We spent today's lecture discussing the notion of diagonalizability in more depth. For instance, we were able to prove (based on a homework problem that will be turned in at the end of the week) that a linear transformation admits an eigenspace precisely when the sum of the geometric multiplicities of its eigenvalues adds up to the dimension of the ambient space. More succinctly, this says a given $T \in \mathcal{L}(V,V)$ is diagonalizable if and only if $$\sum_\lambda \dim(E_\lambda) = \dim(V).$$ We saw this worked out in a specific example where $V$ is a $2$-dimensional space and $T$ acts on a certain basis $\{v_1,v_2\}$ by $$T(v_1) = v_1+v_2 = T(v_2).$$ We saw that not only was this transformation diagonalizable, but we were even able to find a basis so that $\text{Mtx}_{\mathbb{D},\mathbb{D}}(T)$ was diagonal (and we said what that diagonal matrix was). We then tried to push this idea further: to what degree can we actually determine when $\sum_\lambda \dim(E_\lambda) = \dim(V)$? One step in the process is to compute $\dim(E_\lambda)$, and we saw that this can be achieved by computing $\dim(\ker(T-\lambda \text{id}_V)) = \dim(NS(\text{Mtx}_{\mathbb{B},\mathbb{B}}(T)))$ (for whatever basis $\mathbb{B}$ of $V$ you like). The trickier part to implement, it turns out, is knowing precisely why $\lambda$ are eigenvalues for $T$. This is a puzzle that we'll begin to resolve next class.
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Last class period we developed an algorithm for finding out whether a linear transformation is diagonalizable, but it required us to know all the eigenvalues for a matrix. Unfortunately, at this point we don't have an effective way to answer this question. We already know that $\lambda$ is an eigenvalue for $T$ if and only if $\ker(T-\lambda \text{id}_V)$ is nontrivial, and that this is the same as asking whether $NS(\text{Mtx}_{\mathbb{B},\mathbb{B}}(T)-\lambda I)$ is nontrivial. It's also the case that we've got a lot of ways to determine if a square matrix has a nontrivial nullspace (since this is the same thing as saying the matrix is singular), but unfortunately none of them are especially easy to implement if we don't already know the value of $\lambda$ that we're using. So, we need to invent something new.
To move in this direction, we introduced the notion of a determinant function. A determinant function is simply a function $\det:\mathbb{F}_{n \times n} \to \mathbb{F}$ which satisfies two properties: it must be multiplicative (which means that $\det(AB) = \det(A)\det(B)$ for all $A,B \in \mathbb{F}_{n \times n}$), and for all $c \in \mathbb{F}$ it must have $\det(E_{c\text{R}_1}) = c$, where $E_{c\text{R}_1}$ is the matrix we get by scaling row $1$ of $I_n$ by $c$. We commented that such a function exists, and is in fact unique. We then gave the determinant function for $\mathbb{F}_{2 \times 2}$, which is given by $$\det\left(\left[\begin{array}{cc}a&b\\c&d\end{array}\right]\right) = ad-bc.$$ (We checked that this is the determinant function by simply checking it satisfies the properties a determinant function should have.) We then went on to state a theorem that tells us how the determinant acts on elementary matrices. We put off the proof of this latter result to show an important corollary: a matrix $A$ is invertible if and only if $\det(A) \neq 0$.
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We began class by showing some results that follow from our proof that $A$ is invertible if and only if $\det(A) \neq 0$. One was to show that $\det(A)^{-1} = \det(A^{-1})$. The other was to show that similar matrices have the same determinant. We then moved towards proving the result from last time concerning elementary matrices. The goal ultimately is to use this fact, together with row reduction, to give an ``easy" way to calculate the determinant of a matrix. We finished class in the midst of an (overly complex) argument for why the determinant of a diagonal matrix is the product of its diagonal entries.
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With the theory of determinants in hand, today we explored how one can effectively compute determinants. This begins with our result that the determinant of a triangular matrix is simply the product of its diagonal entries. This means that if one reduces a matrix (via elementary row operations) to a triangular matrix, keeping track of the operations used along the way, one can express the determinant of the original matrix in terms of the (easily computed) determinant of the resultant triangular matrix and the (already computed) determinants of the elementary matrices which produced the row reduction. We saw some examples of this in class, and commented that this technique is typically the most efficient way to compute determinants.
There is, however, another way to compute determinants via so-called Laplace expansion. This is a recursive definition for determinant that expresses the determinant of an $n \times n$ matrix as the alternating sum of $n$-many determinants of matrices of size $(n-1) \times (n-1)$. These $(n-1)\times(n-1)$ matrices are the so-called cofactors of the matrix.
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We started class today by exploring the relationship between matrix inverse and the cofactor values from class class. In particular we defined the adjoint of an $n \times n$ matrix, and argued that for any $A \in \mathbb{F}_{n \times n}$ we have $$A\text{Adj}(A) = \text{adj}(A)A = \det(A)I_n.$$ The proof of this result required us to recognize certain quantities as determinants expressed via Laplace expansion. We saw how this formula gave us the formula for inverse of $2 \times 2$ matrices are proved on a homework assignment long ago,
We finished class by seeing how one can use determinants to find the eigenvalues of a given matrix. We defined $\det(A-\lambda I)$ to be the characteristic polynomial of $A$, and argued that the roots of this polynomial are precisely the eigenvalues of $A$. We worked through a specific example. We defined the notion of algebraic multiplicity of an eigenvalue, and stated that the algebraic multiplicity of an eigenvalue is no smaller than the geometric multiplicity of the eigenvalue. In particular, if any eigenvalue has algebraic multiplicity strictly larger than its geometric multiplicity, then the corresponding matrix will fail to be diagonalizable.
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We spent the class today going through some of the applications of linear algebra to our daily technological lives. The slides for today are available here.