# Linear Algebra Questions (basis, subspaces, rank)

Hi I was wondering if anyone could help me with these questions?

For the first question, I wasn't sure how I should approach it and got stuck here...

For the second question, I've reduced the original matrix to the row echelon form. But to find the rank, I would need to reduce it to the RREF. I'm not sure what to do from here.

I would really appreciate if someone could maybe give me a hint as to how to solve these. Thank in advance!

For the second question, assuming your Gaussian elimination is correct, you have rank $4$ for $t\ne1$; for $t=1$ the matrix has rank $2$.
The first question is more interesting. You want to find the kernel of the linear map $$p\mapsto 2p'-(x^2+1)p'''$$ Let's look at the matrix with respect to the standard basis $\{1,x,x^2,x^3\}$; we have \begin{align} 1&\mapsto 0 \\ x&\mapsto 2 \\ x^2&\mapsto 4x \\ x^3&\mapsto -6 \end{align} so the requested matrix is $$\begin{bmatrix} 0 & 2 & 0 & -6 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ The reduced row echelon form is $$\begin{bmatrix} 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}$$ Can you finish? Don't look at the spoiler below.
A basis for the null space of this matrix is$$\left\{\begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix},\begin{bmatrix} 0 \\ 3 \\ 0 \\ 1 \end{bmatrix}\right\}$$so the basis for the subspace is $\{1,3x+x^3\}$