Elementary column operations列基本変形れつきほんへんけい and changes of variables未知数変換みちすうへんかん - Basic Exercises

date2026-06-06document_iddoc_040e466a5de5f96a207c8dfb1b0627b7description列基本変形を右からの基本行列、列空間、階数、未知数変換の観点から確認する演習である。prerequisites列基本変形の基本 / 行基本変形の基本 / 単位行列・零行列・転置の基本type問題演習content_typeexercisestatusactive

mathlinear-algebraexercisecolumn-operationchange-of-variables

data/lecture/math/linear-algebra/elementary-column-operations.lecture.n.md data/lecture/math/linear-algebra/elementary-row-operations.lecture.n.md data/lecture/math/linear-algebra/identity-zero-and-transpose-matrices.lecture.n.md

1Exercise plan

An elementary column operation列基本変形れつきほんへんけい replaces the columns of a matrix行列ぎょうれつ by another generating system. It preserves the column space列空間れつくうかん and rank階数かいすう, but the unknowns未知数みちすう in $A x = b$ no longer have the same meaning.

2Problem 1

For

A = (\begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix}),

perform $C_{2} \leftarrow C_{2} - 2 C_{1}$ . Express this operation as right multiplication by an elementary matrix基本行列きほんぎょうれつ $F$ , and state the relation between the unknowns in $A x = b$ and $A F y = b$ .

2.1Answer example

With

F = (\begin{matrix} 1 & - 2 \\ 0 & 1 \end{matrix}),

we get

A F = (\begin{matrix} 1 & 0 \\ 3 & - 2 \end{matrix}) .

The equation $A F y = b$ means $A (F y) = b$ , so the original unknown is

x = F y = (\begin{matrix} y_{1} - 2 y_{2} \\ y_{2} \end{matrix}) .

2.2Explanation

A column operation preserves the column space列空間れつくうかん, but changes the coordinates of the unknown. When following the equation, record $x = F y$ .

3Problem 2

For the same $A$ , express these column operations as right multiplication by elementary matrices.

$C_{1} \leftrightarrow C_{2}$
$C_{2} \leftarrow 3 C_{2}$

3.1Answer example

Column exchange is represented by

F_{1} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}), A F_{1} = (\begin{matrix} 2 & 1 \\ 4 & 3 \end{matrix}) .

Multiplying a column by a nonzero constant is represented by

F_{2} = (\begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}), A F_{2} = (\begin{matrix} 1 & 6 \\ 3 & 12 \end{matrix}) .

3.2Explanation

Each column of the matrix on the right specifies a new column as a linear combination線型結合せんけいけつごう of the original columns. Therefore column operations are represented by right multiplication.

4Problem 3

A matrix $A$ is changed to $B$ by the elementary row operation行基本変形ぎょうきほんへんけい $R_{2} \leftarrow R_{2} + 3 R_{1}$ . What operation changes $A^{T}$ into $B^{T}$ ?

4.1Answer example

After transposition, the second row operation becomes the column operation

C_{2} \leftarrow C_{2} + 3 C_{1.}

4.2Explanation

Transposition転置てんち interchanges rows and columns, so a row operation becomes the corresponding column operation.

5Problem 4

Point out the error in this explanation:

"Since column operations preserve rank階数かいすう, $A x = b$ and $A F x = b$ have the same solution set解集合かいしゅうごう."

5.1Answer example

Preserving rank means the way the column space列空間れつくうかん is generated has changed, not that the same unknowns solve the equation. $A F x = b$ means $A (F x) = b$ , so one must track a change of variables未知数変換みちすうへんかん such as $z = F x$ .

5.2Explanation

Elementary row operations行基本変形ぎょうきほんへんけい preserve solution sets; elementary column operations列基本変形れつきほんへんけい preserve column space and rank. Similar operations preserve different objects.

6Related exercises

data/exercise/math/linear-algebra/elementary-operations-and-linear-systems.exercise.n.md data/exercise/math/linear-algebra/echelon-forms-and-gaussian-elimination.exercise.n.md

7Related lectures

data/lecture/math/linear-algebra/elementary-column-operations.lecture.n.md data/lecture/math/linear-algebra/elementary-row-operations.lecture.n.md data/lecture/math/linear-algebra/identity-zero-and-transpose-matrices.lecture.n.md

8Proof exercise: column operations preserve column space

8.1Problem

Prove that column operations do not change the column space列空間れつくうかん. Also explain why the solution set解集合かいしゅうごう is not necessarily preserved as-is.

8.2Answer

Swapping columns only changes the order of a generating system. Multiplying a column by a nonzero constant can be undone. In $C_{i} \leftarrow C_{i} + λ C_{j}$ , the new column is a linear combination of the old columns, and the inverse operation exists. Thus the two column spaces contain each other.

However, columns correspond to unknown variables. A column operation changes the meaning of those variables, so the solution set is not preserved without a variable change.

8.3Explanation

Row operations change the equation side; column operations change the generating-system or input-coordinate side.