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Glacius
Linear algebraConcept reference

Dimension

The dimension of a finite-dimensional subspace is the number of vectors in any basis of that subspace.

1 min read1 figureSources & further reading
On this page 7 sections
  1. Overview
  2. Dimension counts the vectors in a basis
  3. One basis for an allowed-output subspace has two patterns
  4. The zero subspace contains only the zero vector
  5. Key takeaway
  6. Sources & further reading
  7. Concept connections

01Dimension counts the vectors in a basis#

Dimension counts the vectors in a basis. It tells you how many independent directions are needed to describe the subspace.

The two vectors shown form a basis of S. Each has three entries, but this subspace has dimension 2.

Two whole basis vectors, (1,0,1) and (0,1,1), are shown in separate cards. They have three entries each, but there are two basis vectors.Two whole basis vectors, (1,0,1) and (0,1,1), are shown in separate cards. They have three entries each, but there are two basis vectors.
Figure 1Two whole basis vectors, (1,0,1) and (0,1,1), are shown in separate cards. They have three entries each, but there are two basis vectors.
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Check your reasoning

B={(1,0,0,0), (0,1,0,0), (0,0,1,1)}B=\{(1,0,0,0),\,(0,1,0,0),\,(0,0,1,1)\} is a basis of S. What is dim⁡S\dim S?

Show answer and explanation
3

Count the 3 whole basis vectors.

02One basis for an allowed-output subspace has two patterns#

One basis for an allowed-output subspace has two patterns. Choosing a different basis for the same subspace changes the patterns, but still needs two. All bases of a fixed subspace have the same size.

Check your reasoning

A basis for an allowed-output subspace S has 2 vectors, each with 4 entries. A different basis describes the same S. How many vectors does it have?

Show answer and explanation
2

Both bases describe S, so both contain 2 vectors.

03The zero subspace contains only the zero vector#

The zero subspace contains only the zero vector. Its basis is the empty list: there are no directions to supply. Its dimension is 0. A list containing the zero vector would be dependent.

Check your reasoning

B={(1,0,1), (0,1,1)}B=\{(1,0,1),\,(0,1,1)\} is a basis of S. Correct the claim dim⁡S=3\dim S=3 (entries per vector).

Show answer and explanation
2

Count the two basis vectors.

Key takeaway

Count basis vectors, not entries per vector. Every basis of the same subspace has the same number of vectors.

  • Determine subspace dimension from a basis.

Sources & further reading

  1. [1]

Reference this concept

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

Glacius. “Dimension.” Math behind ML. /learn/la-dimension