The main topics of this course will be:

(1) Systems of linear equations.
(2) Matrix arithmetic.
(3) Vectors and vector spaces.
(4) Linear transformations.
(5) Determinant.
(6) Orthogonality and least squares.

The main topics covered were:

(1) An equation $F(x,y)=0$ in two variables represent a constraint on the pair $(x,y)$. Solving a *system* of equations $\{F(x,y)=0, G(x,y)=0\}$ involves finding all pairs that satisfy all constraints. Geometrically, this corresponds to finding the intersection points among the solution sets of the individual equations.

(2) What is a linear combination?

(3) What is a system of linear equations?

(4) Elementary row operations = the allowable steps in Gaussian elimination:
  (a) swapping two equations,
  (b) scaling an equation by a nonzero value,
  (c) adding a multiple of one equation to another.

(5) Pivots.

(6) Row echelon form.

(7) Linear systems with real coefficients can have 0 solutions, 1 solution, or infinitely many solutions.

We covered the first 10 slides from the author's first set of slides.

The main topics covered were:

(1) Coefficient matrix of a linear system.

(2) Augmented matrix of a linear system.

(3) $k$th row pivot.

(4) Pivot column, pivot variable, free variable.

(5) (Reduced) row echelon form.

(6) Parametric form of solution set.

This included the following topics:

(1) Matrix notation.

(2) Column vectors and row vectors.

(3) The additive arithmetic of matrices and vectors (addition, negation, scaling, and zero).

(4) We began a discussion of matrix multiplication, noting in particular that if $A$ is $m\times n$, $B$ is $p\times q$, then $AB$ is defined if and only if $n=p$. If $n=p$, then $AB$ is an $m\times q$ matrix.

(5) We proved that matrix addition is commutative and sketched the proofs that matrix addition and multiplication are associative.

(6) We gave an example to show that matrix multiplication need not be commutative.

Then we discussed the following topics:

(1) Matrix multiplication.

(2) Identity matrices.

(3) $2$-sided and $1$-sided inverses.

(4) The axioms of rings.

(5) The fact that $M_n(\mathbb R)$ is a ring.

(6) Each linear system describes a matrix equation of the form $A{\bf x}={\bf b}$.

We took Quiz 1.

(1) Geometric interpretation of vectors in space.

(2) Dot product of two vectors in $\mathbb R^n$.

(3) How to compute the Pythagorean length of a vector in $\mathbb R^n$: $\|{\bf u}\|^2 = {\bf u}\boldsymbol{\cdot} {\bf u}$.

(4) How to compute the angle between vectors in $\mathbb R^n$: $\cos(\theta)={\bf u}\boldsymbol{\cdot}{\bf v}/\|{\bf u}\|\cdot\|{\bf v}\|$.

(5) Geometric interpretation of addition, negation, scaling, and ${\bf 0}$. (Parallelogram rule for addition.)

(6) Linear combinations of vectors.

(7) Definition of ``linear transformation''.

(8) Thm. A function from $\mathbb R^n$ to $\mathbb R^m$ is a linear transformation if and only if it has the form $T({\bf x})=A\cdot {\bf x}$ for some $m\times n$ matrix $A$.

(9) The problem of finding the solutions to $A\cdot {\bf x} = {\bf b}$ is the problem of finding the preimage of ${\bf b}$ under the linear transformation $T({\bf x})=A\cdot {\bf x}$.

(1) Definition of ``span of a set of vectors''.

(2) A solution to $A{\bf x}={\bf b}$ is a vector of coefficients witnessing that ${\bf b}$ belongs to the span of the columns of $A$.

(3) Definition of ``independent set of vectors''.

(1) The intuition behind dimension.

(2) A computational way to determine if a set $V\subseteq \mathbb R^m$ spans $\mathbb R^m$. (Create a matrix $A$ whose columns are the vectors in $V$. Perform Gaussian Elimination on $A$. The span of $V$ is $\mathbb R^m$ if and only if the final row does not consist of zeros. Equivalently, every row of $A$ contains a pivot.)

(3) An algebraic characterization of the property in (2): $\mathbb R^m$ is the span of the columns of $A$ if and only if $A$ has a right inverse.

We took Quiz 2.

First set of statements (equivalent to each other):

(1) $V$ spans $\mathbb R^m$.
(2) The RRE form of $A$ has no zero row.
(3) The RRE form of $A$ has a pivot in every row.
(4) $A{\bf x}={\bf b}$ has at least one solution for every ${\bf b}\in \mathbb R^m$.
(5) $A$ has a right inverse.

Second set of statements (equivalent to each other, but not equivalent to the above five statements):

(1) $V$ is independent.
(2) The RRE form of $A$ has no free variables.
(3) The RRE form of $A$ has a pivot in every column.
(4) $A{\bf x}={\bf b}$ has at most one solution for every ${\bf b}\in \mathbb R^m$.
(5) $A$ has a left inverse.

We defined ``basis'' (= independent spanning set) and ``dimension'' (= size of a basis).

We described algorithms for finding left or right inverses for matrices that have them.

We began a discussion of fields (read pages 153-154) and vector spaces (read pages 83-92).

We took Quiz 3.

We worked through examples to compute the dimension of some spaces and some of their subspaces.

Some of the examples considered were:
(1) If $A$ is an $m\times n$ matrix, then the solution set of the homogeneous equation $A{\bf x}={\bf 0}$ is a subspace of $\mathbb R^n$. The dimension of this subspace equals the number of free variables of the system.
(2) $M_{m\times n}(\mathbb R)$ has dimension $mn$. The collection of symmetric $2\times 2$ matrices ($M^t=M$) over $\mathbb R$ is a subspace of $M_{2\times 2}(\mathbb R)$ of dimension $3$. The collection of antisymmetric $2\times 2$ matrices ($M^t=-M$) over $\mathbb R$ is a subspace of $M_{2\times 2}(\mathbb R)$ of dimension $1$.
(3) The polynomial functions $\{1,x,x^2,x^3,\ldots\}$ form an infinite independent subset of the real vector space $C(\mathbb R)$ of continuous functions $f:\mathbb R\to \mathbb R$. The existence of an infinite independent subset is enough to show that $C(\mathbb R)$ cannot be finite dimensional. (So it is infinite dimensional.)
(4) Let $S$ be the subspace of $C(\mathbb R)$ spanned by $\{\cos(x+r)\;|\;r\in \mathbb R\}$. We used the trig identity $\cos(x+r)=\cos(r)\cos(x)-\sin(r)\sin(x)$ to argue that $\{\cos(x), \sin(x)\}$ is a basis for $S$. Hence $S$ is $2$-dimensional.

Let $\mathbb V$ be a vector space. A subset $G\subseteq \mathbb V$ generates $\mathbb V$ if $\mathbb V = \textrm{span}(G)$. $\mathbb V$ is finitely generated if it has a finite spanning set.

We explained why, if $\mathbb V$ is a finitely generated vector space,

(1) Any spanning subset contains a basis.

(2) Any independent subset can be enlarged to a basis.

(3) All bases have the same cardinality, which we call $\dim(\mathbb V)$.

The same is true even if $\mathbb V$ is not finitely generated, but the proof in the non-finitely generated case requires some background from set theory.

We took Quiz 4.

We reviewed for the midterm exam.

(No new HW assigned for the week of Feb 24-March 3!)

Answers.

We discussed how to write a vector in coordinates relative to a basis. We defined linear transformation (= homomorphism of vector spaces), and isomorphism. We explained why the transpose map is an isomorphism from $M_{2\times 2}(\mathbb R)$ to itself.

We discussed isomorphisms of social networks (as an example of isomorphism between nonalgebraic structures), and then isomorphisms between vector spaces (as an example of isomorphism between algebraic structures). In general, if $\mathbb A$ and $\mathbb B$ are structures of the same type, then an isomorphism from $\mathbb A$ to $\mathbb B$ is
(i) a homomorphism (= structure preserving map) $T\colon \mathbb A\to \mathbb B$, for which there is a function $S\colon \mathbb B\to \mathbb A$ such that
(ii) $S$ is the inverse of $T$: $S(T(a))=a$ and $T(S(b))=b$ for every $a\in \mathbb A$ and $b\in \mathbb B$, and
(iii) $S$ is also a homomorphism.

Using social networks, we gave examples of $T$ and $S$ satisfying (i) and (ii) but not (iii). But we stated that if $\mathbb A$ and $\mathbb B$ are algebraic structures of the same type, and $T$ and $S$ exist satisfying (i) and (ii), then (iii) must also be satisfied. Thus, an isomorphism of vector spaces is an invertible linear transformation. (That is, it is a 1-1, onto linear transformation.)

We then spoke about existence and uniqueness of matrix representations for linear transformations. We discussed the Universal Mapping Property (any set function defined on a basis, mapping into a space, uniquely extends to a linear transformation).

We took Quiz 5.

We explained why matrix multiplication is linear, and why (given appropriate bases) every linear map can be represented by a matrix. We started discussing how to find a change of basis matrix ${}_{\mathcal C}[\textrm{id}]_{\mathcal B}$ from basis $\mathcal B$ to basis $\mathcal C$. The algorithm was to perform Gaussian Elimination on the matrix $[{\mathcal C}|{\mathcal B}]$ to obtain $[I|X]$, and then take ${}_{\mathcal C}[\textrm{id}]_{\mathcal B}$ to be $X=[{\mathcal C}^{-1}]\cdot [{\mathcal B}]$.
Here, when I write an ordered basis ${\mathcal B}$ with square brackets $[{\mathcal B}]$, I mean the matrix whose columns are the vectors in ${\mathcal B}$ written in the correct order.

We discussed the canonical factorization of an arbitrary function $f\colon A\to B$ ($f=\iota\circ \overline{f}\circ \nu$), and discussed related terminology (domain, codomain, image, coimage, natural map, inclusion map, induced map, preimage). We then translated this terminology into the setting of linear transformations. We proved that the image of a linear transformation is a subspace of the codomain of the transformation.

We took Quiz 6.

(1) It is a lattice, $\textrm{Sub}(\mathbb V)$. (The greatest lower bound of $S, T\leq \mathbb V$ is $S\cap T$, the least upper bound of $S, T\leq \mathbb V$ is $S+T=\textrm{span}(S\cup T)$.)

(2) We examined $\textrm{Sub}(\mathbb V)$ when $\mathbb V$ is a $\mathbb F_2$-space of dimension $2$ or $3$.

(3) We explained why $\textrm{Sub}(\mathbb V)$ is graded by dimension.

(4) We explained why $\textrm{Sub}(\mathbb V)$ is complemented. (Given a basis $\mathcal B$ for $\mathbb V$ and a basis $\mathcal C$ for $S$, the column space algorithm applied to $[{\mathcal C}|{\mathcal B}]$ yields a basis for $\mathbb V$ that includes all vectors from $\mathcal C$. Delete the vectors from $\mathcal C$, and what remains will be a basis for a complement $S^{\prime}$ of $S$.)

(5) We stated but did not prove the following dimension formula, which is analogous to the principle of inclusion and exclusion: $\dim(S+T)=\dim(S)+\dim(T)-\dim(S\cap T)$.

(2) We explained why, if we restrict a linear transformation $T\colon \mathbb V\to \mathbb W$ to a subspace $\mathbb S\leq \mathbb V$, then $T|_{\mathbb S}$ is 1-1 iff $\mathbb S\cap \ker(T)=\{0\}$, $T|_{\mathbb S}$ maps $\mathbb S$ onto $\textrm{im}(T)$ iff $\mathbb S + \ker(T)=\mathbb V$, and $T|_{\mathbb S}$ is an isomorphism from $\mathbb S$ to $\textrm{im}(T)$ iff $\mathbb S$ is a complement of $\ker(T)$.

(3) We sketched the proof that $\dim(S+T)=\dim(S)+\dim(T)-\dim(S\cap T)$.

We discussed signed volume and the right hand rule. We explained why signed volume in $\mathbb R^m$ is

(1) a scalar-valued function $f\colon (\mathbb R^m)\times \cdots \times (\mathbb R^m)\to \mathbb R$,

(2) multilinear,

(3) alternating, and

(4) normalized to $1$ on the unit hypercube $({\bf e}_1,\ldots,{\bf e}_m)$.

I claimed that there is a unique function satisfying (1)-(4), but did not prove this yet. We did prove that a multilinear function over a scalar field where $2\neq 0$ is alternating if and only it is multilinear. Thus, over $\mathbb R$, item (3) is equivalent to the slightly weaker property (3)', which says that $f$ is antisymmetric.

We took Quiz 7. Solutions.

We took Quiz 8.

(1) $\det\colon M_{n\times n}(\mathbb F)\to \mathbb F$ is the unique $n$-ary function (of the columns) that is multilinear, alternating, and normalized to $1$ on the standard basis/identity matrix.

(2) If $f\colon M_{n\times n}(\mathbb F)\to \mathbb F$ is any $n$-ary function (of the columns) that is multilinear and alternating, then $$f({\bf v}_1,\ldots,{\bf v}_n)=\det([{\bf v}_1,\ldots,{\bf v}_n])\cdot f({\bf e}_1,\ldots,{\bf e}_n).$$

(3) Any alternating linear function is multilinear. The converse is true if the characteristic of $\mathbb F$ is not $2$.

(4) The determinant may be computed by the permutation expansion, the Laplace expansion, or Gaussian elimination.

(5) $\det(AB)=\det(A)\cdot \det(B)$.

(6) $\det(A)=\det(A^t)$.

(7) $\det(A)\neq 0$ iff the columns of $A$ are independent iff the rows of $A$ are independent.

(8) If $U$ is upper triangular, then $\det(U)$ is the product of the diagonal entries of $U$. (Same type of remark for lower triangular matrices.)

In all of todays discussion, we considered linear transformations $T\colon \mathbb V\to \mathbb V$ from a space to itself. Usually we assumed that $\mathbb V = \mathbb R^n$ and that $T(x) = Ax$ where $A$ is some $n\times n$ matrix. The most important points were:

(1) $v$ is an eigenvector for $A$ if it is a nonzero vector such that there exists a scalar $\lambda$ (lambda) such that $Av=\lambda v$. ($v$ is a ``preserved direction'', or ``axis'' for $T(x)=Ax$.) $\lambda$ is called the eigenvalue for $v$, and $v$ is also called a ``$\lambda$-eigenvector''.

(2) If $A$ is diagonal, then $\mathbb V$ has an ordered basis consisting of eigenvectors for $A$.

(3) Conversely, if $\mathbb V$ has an ordered basis consisting of eigenvectors for $A$, then with respect to that basis the matrix for $A$ will be diagonal.

(4) A nonzero vector $v$ is a $\lambda$-eigenvector for $A$ iff $Av=\lambda v$ iff $(\lambda I - A)v = 0$ iff $v\in \textrm{null}(\lambda I-A)$. That is, $v$ is a nonzero vector in $V_{\lambda}=\textrm{null}(\lambda I-A)$. $V_{\lambda}$ is called the $\lambda$-eigenspace of $A$.

(5) A scalar $\lambda$ is an e-value for $A$ iff $\lambda I-A$ has nontrivial nullity iff $\lambda I-A$ is not of full rank iff $\lambda I-A$ is singular iff $\det(\lambda I-A) = 0$.

(6) The characteristic polynomial of $A$ is defined to be $\chi_A(\lambda) = \det(\lambda I-A)$. (Here $\chi$ is the Greek letter chi, for characteristic, and $\lambda$ is considered to be a variable.) If $A$ is $n\times n$, then the characteristic polynomial of $A$ has degree $n$ and leading coefficient $1$. The e-values of $A$ are the roots of $\chi_A(\lambda)$.

(7) If $A=[a]$, then $\chi_A(\lambda) = \lambda-a$.

(8) If $A$ is $2\times 2$, then $\chi_A(\lambda) = \lambda^2-\textrm{tr}(A)\lambda+\det(A)$.

(1) Eigenspaces for distinct eigenvalues of $A$ are ``disjoint'' (or ``independent'').
(2) If $T(x)=Ax$ has a basis ${\mathcal B}$ of e-vectors, and $C=[{\mathcal B}]$, then $C^{-1}AC = \Lambda$ is the matrix for $A$ with respect to the basis ${\mathcal B}$, and it is a diagonal matrix with diagonal values equal to the e-values for $A$.

We took Quiz 9.

(1) If $\mathcal B$ is a basis for which $D={}_{\mathcal B}A_{\mathcal B}$ is diagonal, then $D$ and $A$ have the same e-values, and they appear as the diagonal entries of $D$.
(2) One may diagonalize $A$ by conjugating it ($C^{-1}AC$) by a matrix $C=[{\mathcal B}]$ where ${\mathcal B}$ is a basis consisting of e-vectors for $A$.
(3) If $D=C^{-1}AC$ is a diagonal form for $A$, then $D^k=C^{-1}A^kC$ and $A^k=CD^kC^{-1}$.
(4) $A$ will fail to be diagonalizable over the scalar field $\mathbb F$ if $\mathbb F$ does not contain all of the e-values of $A$. But this problem may be overcome by extending $\mathbb F$ to its algebraic closure, $\overline{\mathbb F}$.
(5) The other reason that $A$ may fail to be diagonalizable is that the algebraic multiplicity of some e-value of $A$ is strictly larger than the geometric multiplicity of the e-value.

(1) The Cayley-Hamilton Theorem.
(2) Definition of the minimal polynomial of a matrix over a given scalar field ($\textsf{minpoly}_{A,\mathbb F}(\lambda)$).
(3) $A$ is diagonalizable over $\mathbb F$ iff the minimal polynomial of $A$ over $\mathbb F$ factors over $\mathbb F$ into distinct linear factors.
(4) The use of matrices to solve linear recurrences.
(5) The Binet formula for the Fibonacci numbers.

We took Quiz 10.

(1) The structure of an endomorphism of a finite-dimensional vector space.
(2) Similarity.
(3) Algebraically closed fields. The Fundamental Theorem of Algebra.
(4) Similar matrices have the same characteristic polynomials and therefore the same e-values.
(5) The algebraic multiplicity of an e-value is at least as large as the geometric multiplicity.
(6) Distinct e-spaces are independent. Any concatenation of bases for the e-spaces is an independent set, and it is a basis for the space iff the algebraic multiplicity equals the geometric multiplicity for each e-value.

We discussed generalized e-vectors, generalized e-spaces, and Jordan form.

We finished the discussion of the Jordan form.