Need help finding range of arbitrary matrices

Question

I need help with a question about range and column space. So, let's say we have a matrix with non zero vectors a and b in $ R^n $ arranged in such a way that we obtain the matrix A =

$$ \begin{bmatrix} a_1b_1 & a_1b_2 & a_1b_3 & a_1b_4 \\ a_2b_1 & a_2b_2 & a_2b_3 & a_2b_4 \\ a_3b_1 & a_3b_2 & a_3b_3 & a_3b_4 \\ a_4b_1 & a_4b_2 & a_4b_3 & a_4b_4 \\ \end{bmatrix} $$

Now my problem is I am not sure how to find the range of a matrix like this, and here is what I understand about Range/Column Space:

$ R(A) = {Ax ; x \in R^n} $

Range is equal to some linear combination of the columns.

The Range of A is also equal to the orthogonal complement of the Kernel of the transpose of A.

However, I'm not really sure how I can apply any of these to solve this problem. Although, the problem doesn't explicitly state for the range of A but the dimension of that range so, I am starting to think that maybe I just need to find the dimension of the range without having to solve for the range, but wouldn't know how that would work either. Thanks for any help or leads in the right direction, linear algebra seems to be my biggest obstacle yet.

Answer 1

Note that every column of this matrix is a scalar multiple of $\mathbf a = [a_1,a_2,a_3,a_4]^T$, so unless it’s the zero matrix, its column space is simply the span of $\mathbf a$.

Answer 2

The matrix $A$ in your question is the outer product of the vectors $a$ and $b$.

If $a$, $b \in \mathbb{R}^n$, then the outer product is defined as the $n$-by-$n$ matrix $ab^\top $.

Now compute $(ab^\top) x$ and go from there.