Column Space and Row Space

Key idea

Spans of columns and rows of a matrix.

In depth

The column space of A is the span of its columns, equal to all outputs Ax. The row space is the span of its rows. Both have dimension rank(A).

Worked example

For A = [1 2; 2 4], the column space is the line spanned by (1, 2).

Real-world uses

• Finding which outputs a system can produce and which information is redundant in data

Common mistake

Using the columns of the reduced matrix; you need the original columns for the basis.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Linear Algebra

• Triple Product – Scalar triple product gives parallelepiped volume.
• Homogeneous Systems – Ax = 0 always has the trivial solution.
• Cofactor Expansion – Compute determinants recursively.
• Properties of Determinants – Multiplicative; row swaps change sign.
• Geometric Meaning of Determinants – Area or volume scaling factor.
• Cramer’s Rule for Systems – Solve systems using determinants.