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Exercise 6

Avg difficulty 2.5

Suppose that AA is a 3×43\times 4 matrix.

  1. Is it possible that dim⁡ Nul(A)=0\dim~\nul(A) = 0?

  2. If dim⁡ Nul(A)=1\dim~\nul(A) = 1, what can you say about Col(A)\col(A)?

  3. If dim⁡ Nul(A)=2\dim~\nul(A) = 2, what can you say about Col(A)\col(A)?

  4. If dim⁡ Nul(A)=3\dim~\nul(A) = 3, what can you say about Col(A)\col(A)?

  5. If dim⁡ Nul(A)=4\dim~\nul(A) = 4, what can you say about Col(A)\col(A)?

Austin, Understanding Linear Algebra, §3.5 Subspaces, Exercise 6

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Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.