Basics of rank階数かいすう

date2026-07-14document_iddoc_ae771dde9206a080e233af7a5f63cd9adescription行列の階数を列空間の次元として定義し、ピボット・行基本変形・列基本変形・解空間との関係を、線型写像への一般化前の段階として整理する講義である。prerequisites列ベクトルの独立性 / 行基本変形 / 列基本変形 / 階段形と掃き出し法type講義content_typelecturestatusactive

mathlinear-algebraundergraduatelecturerank

1Introduction

In this lecture, the rank階数かいすう of an $m \times n$ matrix行列ぎょうれつ is first defined as the dimension of its column space列空間れつくうかん. The generalized definition for linear maps, as image dimension, is treated in the next lecture.

data/lecture/math/linear-algebra/rank-and-nullity-of-linear-maps.lecture.n.md

2Terms and definitions

Let $A$ be an $m \times n$ matrix行列ぎょうれつ with columns $C_{1}, \dots, C_{n}$ . The column space列空間れつくうかん is

Col (A) = span (C_{1}, \dots, C_{n}) .

The matrix-level definition of rank階数かいすう is

rank (A) = \dim Col (A) .

The row space行空間ぎょうくうかん is the span of the rows. Row rank and column rank will be proved equal. The matrix-level kernel核かく is

\ker A = {x \in K^{n} ∣ A x = 0} .

3Plan

Rank is computed by pivots, but the meaning of the row and column operations must be separated. Row operations preserve rank but usually move the column space. Column operations preserve the column space itself. The number of pivots counts the independent directions that remain in the columns.

4Intuitive explanation

The columns of a matrix are generators in the output space. A column that is a linear combination of previous columns adds no new direction. Rank counts the number of genuinely independent directions spanned by the columns. For a square matrix, failure of full rank means some input direction is collapsed, so no inverse matrix exists.

5Precise explanation

5.11. Why row operations preserve rank

An elementary row operation is left multiplication by an invertible matrix $E$ . Hence

Col (E A) = E (Col (A)) .

Since $E$ is an invertible linear map, it preserves dimension, so $rank (E A) = rank (A)$ . The column space itself may move, so row operations should not be used to read a concrete basis of the original column space without returning to the original pivot columns.

5.22. Why column operations preserve the column space

An elementary column operation is right multiplication by an invertible matrix $F$ . The columns of $A F$ are linear combinations of the columns of $A$ , so $Col (A F) \subset Col (A)$ . Since $A = (A F) F^{- 1}$ , the reverse inclusion also holds. Therefore

Col (A F) = Col (A), rank (A F) = rank (A) .

5.33. Why pivots count rank

Row reduction preserves rank. In echelon form, pivot columns provide the independent directions, and non-pivot columns are linear combinations of pivot columns. Thus rank equals the number of pivots.

5.44. Row rank equals column rank

In echelon form, the number $r$ of nonzero rows equals the number of pivots. The nonzero rows are linearly independent, so row rank is $r$ . The pivot count also gives column rank $r$ . Since row operations preserve both dimensions, row rank and column rank agree for the original matrix.

5.55. Relation to the solution space

For $A x = 0$ , if there are $r$ pivot columns among $n$ variables, then there are $n - r$ free variables. Hence the matrix-level kernel satisfies

\dim \ker A = n - rank (A) .

This is the matrix version of the rank-nullity theorem proved in the next lecture.

6Worked example

For

A = (\begin{matrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 1 & 0 \end{matrix}),

row reduction gives

(\begin{matrix} 1 & 0 & - 3 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \end{matrix}) .

There are two pivots, so $rank (A) = 2$ . To obtain a basis of the original column space, use the original pivot columns, not the reduced columns. Since there are three variables and two pivots, $\dim \ker A = 1$ .

7Generalization to linear maps

This lecture used the matrix-level definition. In the next lecture, rank is redefined for a linear map $T : V \to W$ by

rank (T) = \dim im T, nullity (T) = \dim \ker T .

After bases are chosen, the generalized definition agrees with the matrix definition above.

data/lecture/math/linear-algebra/rank-and-nullity-of-linear-maps.lecture.n.md

8How to recognize what to use

Pivot count is rank. For a basis of the original column space, row-reduce to locate pivot positions but return to the original columns. Column operations preserve the column space itself but change the coordinate reading of unknowns. For square matrices, full rank connects to inverse matrices and determinants.

9Scope

This lecture defines rank for finite matrices. For abstract linear maps, the natural definition is image dimension. Determinants belong to square matrices, but rank is defined for rectangular matrices as well.

10Final forms

[PARSE ERROR: Undefined("Command(\"boxed\")")] rank (A) = \dim Col (A)

[PARSE ERROR: Undefined("Command(\"boxed\")")] rank (E A) = rank (A), rank (A F) = rank (A)

[PARSE ERROR: Undefined("Command(\"boxed\")")] rowrank = columnrank

[PARSE ERROR: Undefined("Command(\"boxed\")")] \dim \ker A = n - rank (A)

11In one sentence

Matrix rank is the dimension of the column space.
Pivots compute that dimension.
The next lecture generalizes rank to image dimension for linear maps.

12Exercise links

data/exercise/math/linear-algebra/vector-spaces-bases-and-rank.exercise.n.md