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

Answer:

No

Step-by-step 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.