Linear Algebra

Let $F$ be a field. Note that the ring $F[x]$ of univariate polynomials over $F$ is an infinite-dminesional vector space over $F$ with a nice (standard) basis: $\{1,x,x^2,x^3,x^4, \dots\}$.

Example 2.1   Let $F_n[x]$ denote the set of polynomials of degree $\le n$. This is an $n+1$-dimensional subspace of $F[x]$. We calculate the matrix representing the map $\frac{d}{dx}:F_n[x]\to F_n[x]$ with respect to the standard basis ${\mathbf{B}}:=\{1,x,\dots,x^n\}$.

$$% latex2html id marker 3479\left[\frac{d}{dx}\right]_{{\ma... ...ts & n \\ 0 & 0 & 0 & 0 & \ldots & 0 \\ \end{array}\right]$$

Example 2.2   Consider the rotation of the Euclidean plane ${\mathbb{E}}^2$ by angle $\alpha$ about the origin. We denote this linear map by $R_\alpha:\mathbb{E}^2\to\mathbb{E}^2$. Choose as the basis a pair of perpendicular unit vectors ${\mathbf{e}}$ and ${\mathbf{f}}$. Then we have that $R_\alpha({\mathbf{e}})=\cos(\alpha){\mathbf{e}}+\sin(\alpha){\mathbf{f}}$, and $R_\alpha({\mathbf{f}})=-\sin(\alpha){\mathbf{e}}+\cos(\alpha){\mathbf{f}}$. Therefore the matrix for $R_\alpha$ in the basis ${\mathbf{e}},{\mathbf{f}}$, is

$$% latex2html id marker 3500[R_\alpha]_{{\mathbf{e}},{\math... ...(\alpha)\\ \sin(\alpha)&\cos(\alpha)\\ \end{array}\right].$$

Example 2.3   Consider the reflection of the Euclidean plane ${\bf E^2}$ through the line which makes an angle $\alpha$ with horizontal ( ${\mathbf{e}}$) and passes through the origin. We denote this linear map by $F_\alpha:\mathbb{E}^2\to\mathbb{E}^2$. Then we have $F_\alpha({\mathbf{e}})=\cos(2 \alpha){\mathbf{e}}+\sin(2 \alpha){\mathbf{f}}$, and $F_\alpha({\mathbf{f}})=\sin(2 \alpha){\mathbf{e}}-\cos(2\alpha){\mathbf{f}}$. Therefore the matrix for $F_\alpha$ in the basis ${\mathbf{e}},{\mathbf{f}}$, is

$$% latex2html id marker 3519[F_\alpha]_{{\mathbf{e}},{\math... ...lpha)\\ \sin(2\alpha)&-\cos(2\alpha)\\ \end{array}\right].$$

Example 2.4   If in the previous example, we had chosen the basis ${\mathbf{u}}$ and ${\mathbf{v}}$, where ${\mathbf{u}}$ is a vector along the line of reflection, and ${\mathbf{v}}$ is perpendicular to it, then the matrix for $F_\alpha$ would be:

$$% latex2html id marker 3532[F_\alpha]_{{\mathbf{u}},{\math... ... \begin{array}{rr} 1 & 0 \\ 0 & -1 \\ \end{array}\right].$$

Definition 2.5   Given a square matrix $A$, define $\mathop{\rm tr}\nolimits (A)$, the trace of $A$ to be the sum of the diagonal entries of $A$.

Remark 2.6   The following observations are special cases of a general principle:

• $\mathop{\rm tr}\nolimits ([F_\alpha]_{{\mathbf{e}}... ...athbf{f}}})=0=\mathop{\rm tr}\nolimits ([F_\alpha]_{{\mathbf{u}},{\mathbf{v}}})$.
• $\det([F_\alpha]_{{\mathbf{e}},{\mathbf{f}}})=-1=\det([F_\alpha]_{{\mathbf{u}},{\mathbf{v}}})$.

Definition 2.7   Let $\{{\mathbf{e_1}},\dots,{\mathbf{e_n}}\}$, and $\{{\mathbf{f_1}},\dots,{\mathbf{f_n}}\}$ be two bases for a vector space $V$. The basis change transformation $S:V\to V$ is the unique (invertible) linear map defined by $S:e_i\mapsto f_i$.

Example 2.8   Let ${\mathbf{x}}$ be a vector in $V$. Then ${\mathbf{x}}=\sum \alpha_i {\mathbf{e}}_i=\sum \beta_i{\mathbf{f}}_i$. Notice that

$$S{\mathbf{x}}=S\sum \alpha_i {\mathbf{e}}_i=\sum \alpha_i S{\mathbf{e}}_i=\sum \alpha_i {\mathbf{f}}_i$$.$$$$

Therefore if

$$% latex2html id marker 3569[{\mathbf{x}}]_{{\mathbf{e}}}= ... ...rray}{c} \alpha_1 \\ \vdots \\ \alpha_n \end{array}\right]$$

$[S{\mathbf{x}}]_{{\mathbf{f}}}=[{\mathbf{x}}]_{\mathbf{e}}$. Also, $[S{\mathbf{x}}]_{{\mathbf{f}}}=[S]_{{\mathbf{f}}}[{\mathbf{x}}]_{{\mathbf{f}}}$ by the following exercise. Thus $[{\mathbf{x}}]_{{\mathbf{f}}}=[S^{-1}]_{{\mathbf{f}}}[{\mathbf{x}}]_{{\mathbf{e}}}$

Exercise 2.9   Let $\{{\mathbf{e_1}},\dots,{\mathbf{e_n}}\}$ be a basis for $V$, and let $\{{\mathbf{f_1}},\dots,{\mathbf{f_n}}\}$ be a basis for $W$. Let ${\mathbf{v}}\in V$, and let $A:V\to W$. Show that $[A]_{{\mathbf{e}},{\mathbf{f}}}[{\mathbf{v}}]_{\mathbf{e}}=[A{\mathbf{v}}]_{{\mathbf{f}}}$.

Exercise 2.10   Let $V,W,Z$ be vector spaces over $F$. Let $\{{\mathbf{e_1}},\dots,{\mathbf{e_n}}\}$ be a basis for $V$, $\{{\mathbf{f_1}},\dots,{\mathbf{f_n}}\}$ be a basis for $W$, $\{{\mathbf{g_1}},\dots,{\mathbf{g_n}}\}$ be a basis for $Z$. Let $A:V\to W$, $B:W\to Z$. Show that $[BA]_{{\mathbf{e}},{\mathbf{g}}}=[B]_{{\mathbf{f}},{\mathbf{g}}}[A]_{{\mathbf{e}},{\mathbf{f}}}$.

Exercise 2.11   If $B$, $C$ are $n\times k$ matrices that $B{\mathbf{x}}=C{\mathbf{x}}$ for all ${\mathbf{x}}\in F^k$ then $B=C$.

Remark 2.12   Let ${\mathbf{e}}$, ${\mathbf{e'}}$ be two bases for $V$ with change of basis map $S: {\mathbf{e}}_i\mapsto {\mathbf{e}}_i'$. Let ${\mathbf{f}}$, ${\mathbf{f'}}$ be two bases for $W$ with change of basis map $T: {\mathbf{f}}_i\mapsto {\mathbf{f}}_i'$. Let $A:V\to W$. We want to compare $[A]_{{\mathbf{e}}{\mathbf{f}}}$ with $[A]_{{\mathbf{e'}}{\mathbf{f'}}}$. Let ${\mathbf{x}}\in V$. We have the following identities:

• $[A{\mathbf{x}}]_{{\mathbf{f}}}=[A]_{{\mathbf{e}}{\mathbf{f}}}[{\mathbf{x}}]_{{\mathbf{e}}}$
• $[A{\mathbf{x}}]_{{\mathbf{f'}}}=[A]_{{\mathbf{e'}}{\mathbf{f'}}}[{\mathbf{x}}]... ...\mathbf{f'}}}[S^{-1}]_{{\mathbf{e'}}{\mathbf{e'}}}[{\mathbf{x}}]_{{\mathbf{e}}}$
• $[A{\mathbf{x}}]_{{\mathbf{f'}}}=[T^{-1}]_{{\mathbf{f'}}{\mathbf{f'}}}[A{\mathbf{x}}]_{{\mathbf{f}}}$.

Therefore for all ${\mathbf{x}}\in V$,

$$([T^{-1}]_{{\mathbf{f'}}{\mathbf{f'}}}[A]_{{\mathbf{e}}{\mathbf{f... ...thbf{f'}}}[S^{-1}]_{{\mathbf{e'}}{\mathbf{e'}}})[{\mathbf{x}}]_{{\mathbf{e}}}.$$

However, by the previous exercise, this is only possible if

$$[T^{-1}]_{{\mathbf{f'}}{\mathbf{f'}}}[A]_{{\mathbf{e}}{\mathbf{f}}}=[A]_{{\mathbf{e'}}{\mathbf{f'}}}[S^{-1}]_{{\mathbf{e'}}{\mathbf{e'}}}.$$

Remark 2.13   From the previous remark, we deduce that

$$[A]_{{\mathbf{e}}{\mathbf{f}}}=[T]_{{\mathbf{f'}}{\mathbf{f'}}}[A... ...f{f'}}}[S^{-1}]_{{\mathbf{e'}}{\mathbf{e'}}}=[TAS]_{{\mathbf{e'}}{\mathbf{f'}}}$$.$$$$

Corollary 2.14   Let $S=A=T$. Then

$$[S]_{{\mathbf{e}}{\mathbf{e}}}=[TAS^{-1}]_{{\mathbf{f}}{\mathbf{f}}}=[S]_{{\mathbf{f}}{\mathbf{f}}}$$.$$$$

Corollary 2.15   If $V=W$, $e=f$, then $A_{\text{new}}=S^{-1}A_{\text{old}}.S$

Definition 2.16   Two $n\times n$ matrices $A$, $B$ are similar if there exists some invertible $S$ such that $B=S^{-1}AS$. We write $A\sim B$.

Definition 2.17   The characteristic polynomial $f_A$ of a matrix $A$ is the determinant $f_A(x) =$   det$(xI-A)$, where $I$ is the $n\times n$ identity matrix.

Exercise 2.18   If $A$ and $B$ are $n\times n$ matrices, then det$(AB)=$det$(A)$det$(B)$.

Exercise 2.19   If $A\sim B$ then det$(A)=$det$(B)$.

Exercise 2.20   If $A\sim B$ then $f_A=f_B$.

Definition 2.21   If $A$ is a matrix, the roots of $f_A$ are called the eigenvalues of $A$.

Definition 2.22   If $T$ is a linear transformation, then the characteristic polynomial of $T$ is the characteristic polynomial of a matrix for $T$ in some basis. The last exercise shows that this is well-defined.

Definition 2.23   If $T$ is a linear transformation, then the eigenvalues of $T$ are the roots of the characteristic polynomial.

Remark 2.24   If $A$ is an upper triangular matrix, then the eigenvalues of $A$ are the diagonal entries of $A$, taken with multiplicity.

Exercise 2.25   Find two $2\times 2$ matrices, $A$ and $B$ such that $f_A=f_B$ but $A$ and $B$ are not similar.

Exercise 2.26   If $A$ is an $n\times n$ matrix over a field $F$, and $f_A$ has $n$ distinct roots in $F$, then $A$ is ``diagonalizable,'' i.e., $A$ is similar to a diagonal matrix. (What are the entries of this diagonal matrix?)