Question

3. Could a 5 x 8 matrix have dim Col A 6 and dim Nul A - 2? Justify your answer. (Hint: Think about the possible pivot rows and columns

Answer 1

No

Explanation:

A 5x8 matrix
`A` represents a linear transformation between the vector space
`\mathbb{R}^8` and
`\mathbb{R}^5`. Remeber that the columns of
`A` generate the image of the linear transformation, which is a vector subspace of
`\mathbb{R}^5`. Therefore,
`\dim \text{Col}A \leq 5`, which is the dimension of the the total space.