Which of the following are true?

Question

I need to find which of the following are true?

1. $\exists A\in M_{2\times 5}(\mathbb{R})\ni\dim$ of null space of $A$ is $2$

My ans: False as $\dim Null(A)+\dim Im(A)=5\Rightarrow\dim Im(A)=5-2=3(\Leftrightarrow)$

2. $\exists A\in M_{2\times 5}(\mathbb{R})\ni\dim$ of null space of $A$ is $0$

My ans: False as $\dim Null(A)+\dim Im(A)=5\Rightarrow\dim Im(A)=5-0=5(\Leftrightarrow)$

3. $\exists A\in M_{2\times 5}(\mathbb{R}),\exists B\in M_{5\times 2}(\mathbb{R})\ni AB=I_2$

my ans: True. suppose, $T_1(x_1,\dots,x_5)=(x_1,x_2)$ and $T_2(x_1,x_2)=(x_1,x_2,0,0,0)$ then the matrix of $T_1$ is $A$ and matrix of $T_2$ is $B$ say. $AB$ is $I_2$ which I have calculated.

4. $\exists A\in M_{2\times 5}(\mathbb{R})$ whose null space is $\{(x_1,\dots,x_5):x_1=x_2,x_3=x_4=x_5\}$

I have no idea about this one.

Thanks for correcting me and hlping.

Answer 1

Everything is correct.

In the third one, I think it is a degree of complexity simpler to just take $A=\begin{bmatrix}1 & 0 & 0 & 0 & 0\\0 & 1 & 0 & 0 & 0\end{bmatrix}$ and $B=A^T$-

For the fourth one find a basis of the given subspace, find its dimension and compare to a previous question.