Basics of elementary column operations列基本変形れつきほんへんけい

1Introduction

The key point is that an elementary column operation列基本変形れつきほんへんけい is an operation that reversibly replaces the way the columns of a matrix行列ぎょうれつ are displayed.

Because this page comes before the formal rank lecture, we separate the early matrix-level definitions from the later generalized definitions. Here the column space列空間れつくうかん means the space spanned by the columns, and rank階数かいすう means its dimension. Later, rank is generalized as the dimension of the image像ぞう of a linear map線型写像せんけいしゃぞう.

When this page uses kernel核かく, first read it in the matrix-level sense: the set of input vectors $x$x satisfying $Ax=0$Ax=0. Later, it is defined for a linear map線型写像せんけいしゃぞう $T$T as the set of inputs satisfying $T\left(x\right)=0$T(x)=0.

For a matrix

then column operations reorganize the generating set $a_1,\text{[PARSE ERROR: Undefined("Command("dots")")]},a_n$a_1,\dots,a_n of the column space列空間れつくうかん. They do not merely move symbols around; they change the coordinates on the input side.

2Terms and definitions

The three elementary column operations列基本変形れつきほんへんけい are the following.

The condition $\lambda \ne 0$\lambda\ne0 is necessary in the second operation. If $\lambda =0$\lambda=0, the column itself disappears and cannot be recovered by an inverse operation.

3Why these three are called elementary

Each of the three operations is reversible.

The swap

is undone by swapping the same two columns again.

The scaling operation

is undone by

The replacement operation

is undone by

The condition $\lambda \ne 0$\lambda\ne0 is essential; multiplying a column by 0 destroys information and cannot be undone.

4Strategy

When studying column operations, write the matrix as

Then track how each new column is made as a linear combination of the old columns. The guiding rule is

Indeed, the $k$k-th column of $AF$AF is obtained by using the $k$k-th column of $F$F as the coefficient vector for a linear combination of $a_1,\text{[PARSE ERROR: Undefined("Command("dots")")]},a_n$a_1,\dots,a_n.

5Intuitive explanation

A column operation replaces a list of column generators by another list that spans the same space. For example, after the operation $C_3\leftarrow C_3-3C_1$C_3\leftarrow C_3-3C_1, the old column can be recovered because

Thus the generated space is unchanged, even though the displayed columns have changed.

In the map analogy, a column operation is like changing the signs that describe the same place. The displayed columns change, but the space reachable from them does not.

6Precise explanation

6.11. Multiplying by an elementary matrix on the right

An elementary matrix is obtained by applying one elementary operation to the identity matrix. An elementary column operation can be represented as multiplication by such an elementary matrix on the right.

Consider the operation

It is represented by right multiplication with

Indeed,

The entry $-3$-3 appears in row 1, column 3 of $F$F because the third column of $AF$AF is determined by the third column of $F$F as a coefficient vector.

Column operations therefore have the form

where $F$F is an invertible elementary matrix.

6.22. Meaning as a linear map

Suppose $A$A represents $T_A:K^n\to K^m$T_A:K^n\to K^m and $F$F represents an invertible map $T_F:K^n\to K^n$T_F:K^n\to K^n. Then $AF$AF represents

because for every input $y$y,

If $F$F is the elementary matrix corresponding to an elementary column operation, then $F$F is invertible. Hence $T_F$T_F maps $K^n$K^n onto all of $K^n$K^n, and

Thus the column space and rank are preserved.

The kernel is not generally preserved as the same set. Since $AFy=0$AFy=0 means $A\left(Fy\right)=0$A(Fy)=0,

So the dimension of the collapsed directions is preserved, but their coordinate description changes.

6.33. Difference from elementary row operations

For systems, this distinction matters. If we solve

then the original unknown is not $y$y. It is

Thus column operations can be used in equations only if this change of variables is tracked.

7Concrete examples

Let

7.1Swapping columns

For

we get

7.2Multiplying a column by a nonzero scalar

For

we get

7.3Adding a multiple of another column

For

compute

Thus

7.4Concrete example of a change of variables

Suppose a two-column coefficient matrix undergoes

Then

If we solve

then the original unknown is

So $y$y is the coordinate vector after the column change, not the original unknown vector.

8What is preserved

Because each new column is a linear combination of the old columns and the old columns can be recovered by the inverse operation,

Therefore

The column space列空間れつくうかん and rank階数かいすう are preserved. The kernel核かく is not generally preserved as the same set of coordinate vectors. Since

we have

The dimension of the kernel is preserved, but its coordinate representation changes.

9Effects on determinants

This section is a later-reference supplement. Return to it after the determinant lectures. The main flow of the present page uses only column spaces, rank, and changes of unknowns.

For square matrices, column operations affect the determinant行列式ぎょうれつしき as follows.

The reason is that the determinant is multilinear in the columns and becomes zero when two columns coincide.

10What changes and what is preserved by operation

Writing an elementary column operation列基本変形れつきほんへんけい as $A\mapsto AF$A\mapsto AF, the matrix $F$F is invertible可逆かぎゃく. Since multiplication is on the right, each new column is a linear combination線型結合せんけいけつごう of the original columns.

This explains why

and why rank階数かいすう, the dimension of the column space, is preserved.

The kernel核かく is not generally preserved as the same set. From

putting $x=Fy$x=Fy gives

Thus the dimension of the kernel is preserved, but its coordinate representation changes.

This matters for a system of linear equations連立一次方程式れんりついちじほうていしき. If the coefficient matrix in $Ax=b$Ax=b is changed to $AF$AF, then one is not solving for the same unknown vector $x$x. After solving

the original unknown is recovered by

Column operations can be used to organize equations only if this variable change is tracked. Without tracking, they do not preserve the solution set in the same way that row operations do.

11Criteria

• Use elementary row operations行基本変形ぎょうきほんへんけい when the solution set of a system should be preserved directly.
• Use elementary column operations列基本変形れつきほんへんけい when the column space列空間れつくうかん, rank階数かいすう, or generating set of columns is the focus.
• For determinant computations, either row operations or column operations may be used, but signs and scalar factors must be recorded.
• If a column operation is used while solving $Ax=b$Ax=b, track the change of variables $x=Fy$x=Fy until the end.

12Scope of validity

Elementary column operations列基本変形れつきほんへんけい are defined for arbitrary $m\times n$m\times n matrices. They preserve the column space列空間れつくうかん and rank階数かいすう. Statements about determinant effects apply only to square matrices, because determinants are defined only there.

A column operation replaces the generating set of columns reversibly and keeps the column space in the codomain. On the other hand, the input-side coordinates change, so the unknowns in an equation must not be identified without tracking that coordinate change.

13Final forms

14In one sentence

An elementary column operation列基本変形れつきほんへんけい changes the way column vectors are displayed without changing the space spanned by those columns.

An elementary column operation列基本変形れつきほんへんけい is right multiplication by an invertible elementary matrix. It changes the coordinate description on the input side, so it should be read as a change of variables未知数変換みちすうへんかん, not as the same operation as an elementary row operation.

The preservation rules are the practical checkpoint. Because $AE$AE has columns that are invertible linear combinations of the columns of $A$A, the column space and rank are preserved. For a system $Ax=b$Ax=b, the literal solution vector in the original variables changes, but the transformed system $AEy=b$AEy=b is equivalent after the substitution $x=Ey$x=Ey. Determinants change according to the same swap, scaling, and column-addition rules as row operations.

15Theorem and proof: elementary column operations preserve column space

Elementary column operations列基本変形れつきほんへんけい do not change the space spanned by the columns.

Proof. Write $A=[C_1\cdots C_n]$A=[C_1\ \cdots\ C_n]. Swapping columns only changes the order of a generating set, so the span is unchanged. Multiplying a column by a nonzero scalar only replaces a generator by a nonzero scalar multiple in the same direction, and it can be undone by multiplying by $1/\lambda$1/\lambda.

Finally, consider $C_i\leftarrow C_i+\lambda C_j$C_i\leftarrow C_i+\lambda C_j. The new $C_i$C_i is a linear combination of the old columns, so the span of the new columns is contained in the old column space. The inverse operation $C_i\leftarrow C_i-\lambda C_j$C_i\leftarrow C_i-\lambda C_j recovers the old columns from the new columns, so the reverse inclusion also holds. Therefore the column spaces are equal.

However, column operations change the meaning of the unknown coordinates, so they do not necessarily preserve the solution set of a system of linear equations as the same set of unknown vectors.

16Exercise links

17Related links