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Glacius
Linear algebraConcept reference

Column space

The column space of a matrix consists of all linear combinations of its columns.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. The column space of A contains every output Ax: all combinations of A’s columns
  3. A closer look
  4. A closer look
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01The column space of A contains every output Ax: all combinations of A’s columns#

The column space of A contains every output Ax: all combinations of A’s columns. A basis keeps enough independent columns to produce those same outputs.

For this A, elimination gives pivot columns 1 and 3. Keep (1,2) and (0,1) from the original matrix. Column 2 is twice column 1.

Matrix A has rows (1,2,0) and (2,4,1). The selected original columns 1 and 3 are (1,2) and (0,1). Column 2 is twice column 1, so the selected columns form a basis of the column space.Matrix A has rows (1,2,0) and (2,4,1). The selected original columns 1 and 3 are (1,2) and (0,1). Column 2 is twice column 1, so the selected columns form a basis of the column space.
Figure 1Matrix A has rows (1,2,0) and (2,4,1). The selected original columns 1 and 3 are (1,2) and (0,1). Column 2 is twice column 1, so the selected columns form a basis of the column space.
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Check your reasoning

Select original columns 1 and 2. A=[102215]A=\begin{bmatrix}1&0&2\\2&1&5\end{bmatrix}

  1. A{(1,2), (0,1)}\{(1,2),\,(0,1)\}
  2. B{(1,2), (2,5)}\{(1,2),\,(2,5)\}
  3. C{(1,0), (0,1)}\{(1,0),\,(0,1)\}
Show answer and explanation
{(1,2), (0,1)}\{(1,2),\,(0,1)\}

Use columns 1 and 2 from A.

02A closer look#

Row reduction preserves column-dependence relations, so pivot positions identify which columns to keep. It can change the column vectors themselves. Use the positions from reduction and the values from A.

Check your reasoning

RREF column 1 was copied. Choose the original pivot vector. A=[1236]A=\begin{bmatrix}1&2\\3&6\end{bmatrix}

  1. A(1,0)(1,0)
  2. B(0,3)(0,3)
  3. C(1,3)(1,3)
Show answer and explanation
(1,3)(1,3)

Use (1,3) from A.

03A closer look#

If columns P, Q, R are output patterns, reduction tells you which pattern indices are independent. The selected original patterns span every attainable output; the reduced vectors need not be the same patterns.

Check your reasoning

P, Q, R reduce to E. Original basis columns? E=[120001]E=\begin{bmatrix}1&2&0\\0&0&1\end{bmatrix}

  1. AQ only
  2. BP and Q
  3. CP and R
Show answer and explanation
P and R

Pivots: 1 and 3.

Key takeaway

Find pivot positions by row reduction, then take those columns from the original matrix. Reduced columns can span a different space.

  • Find a basis for the column space of a matrix.

Sources & further reading

  1. [1]

Reference this concept

Link to this page, a section, or an individual figure.

Glacius. “Column space.” Math behind ML. /learn/la-column-space