Activity 6.2.5

Suppose that $W$ is a $5$-dimensional subspace of $\real^9$ and that $A$ is a matrix whose columns form a basis for $W$; that is, $\col(A) = W$.

1. What is the shape of $A$?

2. What is the rank of $A$?

3. What is the shape of $A^T$?

4. What is the rank of $A^T$?

5. What is $\dim\nul(A^T)$?

6. What is $\dim W^\perp$?

7. How are the dimensions of $W$ and $W^\perp$ related?

Suppose that $W$ is a subspace of $\real^4$ having basis $$\wvec_1 = \fourvec102{-1},\hspace{24pt} \wvec_2 = \fourvec{-1}2{-6}3.$$

1. Find the dimensions $\dim W$ and $\dim W^\perp$.

2. Find a basis for $W^\perp$. It may be helpful to know that the Sage command A.right_kernel() produces a basis for $\nul(A)$.

3. Verify that each of the basis vectors you found for $W^\perp$ are orthogonal to the basis vectors for $W$.