# Linear Algebra: Finding the matrix representation with respect to standard basis

I would appreciate some help with a linear algebra practice question, I'm studying for my final and I am stuck, this is a screenshot of the question:

Are my answers correct?

a)

$P_{2}$:

$\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}$

$P_{1}$:

$\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}$

b)

Rank: 2 (as I have two pivots)

Nullity: 0 (as dim - rank= 2 - 2= 0)

• No, the kernel of your map contains $1$. In general if $A:V\to W$ we have $\dim\ker A+\dim{\rm im} A=\dim V$. In your case $\dim V=3$, and $\dim{\rm im}\; A=2$. – Pedro Tamaroff Dec 12 '13 at 21:37
• Meaning your nullity cannot be 0. – Vladhagen Dec 12 '13 at 21:38
• You have to find one matrix $T$ so why do you write two matrices? The rank nullity theorem: $\dim P_2=3=\dim\ker T+\mathrm{rank}(T)=1+2$ – user63181 Dec 12 '13 at 21:40
• a) The answer must be a $2\times 3$ matrix: you are going from $3$ dimensions to $2$ dimensions. – Julien Dec 12 '13 at 21:40

Let $T_A$ be the matrix: $\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$