Find Orthonormal basis for the following vector spaces:
A) The range of the linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^4$ which sends $x \to Ax$ where $A$ is: \begin{bmatrix}1&0\\1&1\\1&1\\0&1\end{bmatrix}
B) The nullspace of the linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^4$ which sends $x \to Ax$ where $A$ is: \begin{bmatrix}1&0\\1&1\\1&1\\0&1\end{bmatrix}
I understand that for part B, the null space can consist of only the zero vector. I know how to complete the magnitude part of the equation, I am just stuck getting to that point.
1) The column of $A$ are linearly independent. Then, the range is just the span of of columns of $A$.
2) The nullspace of $A$ is formed by all vectors \begin{bmatrix}x\\y\end{bmatrix} such that $Ax = 0.$ This turns to be a system of equations:
$$\begin{cases} x = 0\\ x+y = 0\\ y = 0 \end{cases} \Rightarrow \begin{cases} x = 0\\ y = 0 \end{cases}$$
Then, the nullspace contains only the zero vector.