Matrix calculation行列計算ぎょうれつけいさん and linear transformations線型変換せんけいへんかん - Basic Exercises

1Exercise plan

Before computing with a matrix行列ぎょうれつ, check the size conditionサイズ条件じょうけん. The product $AB$AB means "first $B$B, then $A$A," so changing the order順序じゅんじょ changes the meaning and often the result.

Before multiplying matrices, check dimensions and meaning. The product $AB$AB means first applying $B$B and then applying $A$A, so the order is part of the statement. The columns of a matrix are the images of the standard basis vectors, which is often the fastest way to identify the associated linear transformation.

For inverse and transpose questions, keep the structural rules visible: $(AB{)}^{-1}=B^{-1}{A}^{-1}$(AB)^{-1}=B^{-1}A^{-1}, $(AB{)}^T=B^T{A}^T$(AB)^T=B^TA^T, and a map is invertible only when it is one-to-one and onto in the square finite-dimensional setting. If a calculation fails by dimension mismatch, that is not a minor arithmetic error; it means the proposed composition is not defined.

Order note: the detailed computation of inverse matrices is treated later. In this exercise, we use only the definition: for a square matrix $$A$$A, an inverse matrix逆行列ぎゃくぎょうれつ is a matrix satisfying $$A{A}^{-1}=A^{-1}A=I$$AA^{-1}=A^{-1}A=I.

2Check problem: linearity線型性せんけいせい

Define $T:R^2\to R^2$T:\mathbb R^2\to\mathbb R^2 by

Show linearity by checking additivity and homogeneity.

2.1Answer example

For $u=(x_1,y_1{)}^T$u=(x_1,y_1)^T and $v=(x_2,y_2{)}^T$v=(x_2,y_2)^T,

For a scalar $c$c,

Therefore $T$T is a linear map線型写像せんけいしゃぞう.

2.2Explanation

The proof uses arbitrary vectors and scalars, not a few examples.

The additivity check compares the two full expressions

and

for arbitrary $u=(x_1,y_1{)}^T$u=(x_1,y_1)^T and $v=(x_2,y_2{)}^T$v=(x_2,y_2)^T, and the two components agree term by term. The homogeneity check also uses an arbitrary scalar $c$c. This is why the argument is a proof from the definition定義ていぎ, rather than verification on a few numerical examples.

3Problem 1

For

find $A+B$A+B and $2A-B$2A-B, and explain why they are defined.

3.1Answer example

Both are $2\times 3$2\times3 matrices, so addition and subtraction are defined.

3.2Explanation

Matrix sums are entrywise, so row and column counts must match.

4Problem 2

For

compute $AB$AB and $BA$BA and explain the difference.

4.1Answer example

Thus $AB\ne BA$AB\ne BA.

4.2Explanation

$AB$AB applies $B$B first and then $A$A. $BA$BA reverses the order.

The product $AB$AB means that the transformation represented by $B$B is applied first and the transformation represented by $A$A is applied second. Reversing the order changes the geometry: shearing and stretching in different orders generally give different outputs.

5Problem 3

A linear transformation $T:R^2\to R^3$T:\mathbb R^2\to\mathbb R^3 satisfies

Find its representation matrix表現行列ひょうげんぎょうれつ and compute $T(4,-2{)}^T$T(4,-2)^T.

5.1Answer example

The representation matrix is

Then

5.2Explanation

The columns of a matrix are the images of the standard basis vectors, and matrix multiplication is a linear combination of columns.

6Problem 4

Let $A$A be $2\times 3$2\times3 and $B$B be $3\times 4$3\times4. Decide whether $AB$AB, $BA$BA, $A^{T}B$A^TB, and $B{A}^T$BA^T are defined.

6.1Answer example

$AB$AB is defined and has size $2\times 4$2\times4. $BA$BA is not defined. Since $A^T$A^T is $3\times 2$3\times2, $A^{T}B$A^TB is not defined. $B{A}^T$BA^T is also not defined.

6.2Explanation

For a product, the number of columns of the left matrix must equal the number of rows of the right matrix. Recheck sizes after transposition.

7Problem 5

Let $A$A be $m\times n$m\times n. Explain $A{I}_n=A$AI_n=A and $I_{m}A=A$I_mA=A from sizes and transformations. Also state what $A0$A0 and $0A$0A represent.

7.1Answer example

$I_n$I_n is the identity on $R^n$\mathbb R^n and $I_m$I_m is the identity on $R^m$\mathbb R^m. Thus $A{I}_n$AI_n does nothing before $A$A, and $I_{m}A$I_mA does nothing after $A$A. Both equal $A$A.

Multiplying by a zero matrix of suitable size sends everything to zero.

7.2Explanation

The identity matrix preserves information; the zero matrix collapses all input differences. Therefore composing with an identity matrix leaves the transformation unchanged, while composing with a zero matrix can drastically change later invariants such as rank and kernel. Rank and kernel are later vocabulary here; in this problem, the main distinction is that identity keeps the transformation and zero destroys input information.

8Supplementary problems: operations and transpose

8.1Problem 6

Why is the sum of two $2\times 3$2\times3 matrices defined?

8.2Answer example

Matrix addition is entrywise, so the row and column counts must match; both are $2\times 3$2\times3.

8.3Problem 7

For $A=\left(\begin{array}{ccc}1& 0& 2\\ 3& -1& 4\end{array}\right)$A=\begin{pmatrix}1&0&2\\3&-1&4\end{pmatrix}, compute $-2A$-2A.

8.4Answer example

8.5Problem 8

Compare the sizes of $\left(\begin{array}{cc}1& 2\end{array}\right)$\begin{pmatrix}1&2\end{pmatrix} and $\left(\begin{array}{c}1\\ 2\end{array}\right)$\begin{pmatrix}1\\2\end{pmatrix}.

8.6Answer example

The first is $1\times 2$1\times2 and the second is $2\times 1$2\times1.

8.7Problem 9

What is the size of the product of a $2\times 3$2\times3 matrix and a $3\times 4$3\times4 matrix?

8.8Answer example

The product is defined and has size $2\times 4$2\times4.

8.9Problem 10

For $A=\left(\begin{array}{cc}1& 0\\ 2& 1\end{array}\right)$A=\begin{pmatrix}1&0\\2&1\end{pmatrix} and $x=(3,4{)}^T$x=(3,4)^T, compute $Ax$Ax.

8.10Answer example

8.11Problem 11

Give an example where matrix multiplication is not commutative可換かかん.

8.12Answer example

For $A=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$A=\begin{pmatrix}1&1\\0&1\end{pmatrix} and $B=\left(\begin{array}{cc}2& 0\\ 0& 1\end{array}\right)$B=\begin{pmatrix}2&0\\0&1\end{pmatrix},

8.13Problem 12

Find $A^T$A^T for $A=\left(\begin{array}{cc}0& 1\\ 2& 3\end{array}\right)$A=\begin{pmatrix}0&1\\2&3\end{pmatrix}.

8.14Answer example

8.15Problem 13

For $I_{2}A=A$I_2A=A to hold, what number of rows must $A$A have?

8.16Answer example

Since $I_2$I_2 is $2\times 2$2\times2, $A$A must have 2 rows.

8.17Problem 14

Why does order reverse in $(AB{)}^T=B^T{A}^T$(AB)^T=B^TA^T?

8.18Answer example

$AB$AB means first $B$B and then $A$A. Transposition swaps row and column roles, so the composition is read back in the reversed order.

8.19Problem 15

Preview note: rank, row space, and column space are developed in the later rank lectures and exercises. Read this problem as an advanced check of how transpose exchanges rows and columns.

Explain why $\mathrm{rank}\left(A\right)=\mathrm{rank}\left(A^T\right)$\operatorname{rank}(A)=\operatorname{rank}(A^T).

8.20Answer example

The row space of $A$A corresponds to the column space of $A^T$A^T, and row rank equals column rank. Therefore $\mathrm{rank}\left(A\right)=\mathrm{rank}\left(A^T\right)$\operatorname{rank}(A)=\operatorname{rank}(A^T).

8.21Problem 16

Preview note: elementary row operations行基本変形ぎょうきほんへんけい and elementary column operations列基本変形れつきほんへんけい are treated systematically in later exercises on elementary operations. This problem only checks that transpose転置てんち converts a row operation into the corresponding column operation.

For $A=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)$A=\begin{pmatrix}1&2\\3&4\end{pmatrix}, apply $R_2\leftarrow R_2+3R_1$R_2\leftarrow R_2+3R_1 to get $B$B. What column operation changes $A^T$A^T into $B^T$B^T?

8.22Answer example

Since $A^T=\left(\begin{array}{cc}1& 3\\ 2& 4\end{array}\right)$A^T=\begin{pmatrix}1&3\\2&4\end{pmatrix}, the operation is $C_2\leftarrow C_2+3C_1$C_2\leftarrow C_2+3C_1.

8.23Problem 17

For

compute $A\left(Bx\right)$A(Bx) and $\left(AB\right)x$(AB)x.

8.24Answer example

$Bx=\left(\begin{array}{c}6\\ 7\end{array}\right)$Bx=\begin{pmatrix}6\\7\end{pmatrix}, so $A\left(Bx\right)=\left(\begin{array}{c}13\\ 14\end{array}\right)$A(Bx)=\begin{pmatrix}13\\14\end{pmatrix}. Also $AB=\left(\begin{array}{cc}3& 1\\ 2& 2\end{array}\right)$AB=\begin{pmatrix}3&1\\2&2\end{pmatrix}, so $\left(AB\right)x=\left(\begin{array}{c}13\\ 14\end{array}\right)$(AB)x=\begin{pmatrix}13\\14\end{pmatrix}.

8.25Problem 18

For

confirm that the columns of $AB$AB are $A{b}_1,A{b}_2$Ab_1,Ab_2.

8.26Answer example

Thus $AB=\left[\begin{array}{cc}A{b}_1& A{b}_2\end{array}\right]$AB=\begin{bmatrix}Ab_1&Ab_2\end{bmatrix}.

9Related lectures

10Proof exercise: matrix products and inverse matrices

10.1Problem

Prove $\left(AB\right)x=A\left(Bx\right)$(AB)x=A(Bx) using the column viewpoint. Also prove that an inverse matrix, if it exists, is unique.

10.2Answer

Let the columns of $B$B be $b_1,\dots ,b_p$b_1,\ldots,b_p. Then $Bx=\sum x_j{b}_j$Bx=\sum x_jb_j. By linearity,

The $j$j-th column of $AB$AB is $A{b}_j$Ab_j, so $\left(AB\right)x$(AB)x gives the same expression.

If $B,C$B,C are both inverses of $A$A, then

10.3Explanation

Matrix product is composition of linear maps, and an inverse matrix is the unique map that undoes the composition.