Let $T : R^3 → R^4$ be the linear transformation defined by:
$$T(x_1, x_2, x_3) = (3x_1 − 4x_2 − x_3, x_1 + 2x_2 + 3x_3, 6x_1 − x_2 + 5x_3, 10x_2 + 10x_3)$$
Find a basis for the image of $T$, $\operatorname{Im}(T)$, and determine $\dim(\operatorname{Im}(T))$.
Step 1: Standard matrix: $$\begin{bmatrix}3&-4&-1\\1&2&3\\6&-1&5\\0&10&10\end{bmatrix}$$
Step 2: RREF of the standard matrix:
$$\begin{bmatrix}1&0&1\\0&1&1\\0&0&0\\0&0&0\end{bmatrix}$$
Step 3:
As leading entries are in col 1,2 $\operatorname{Im}(T)$ has a basis $\{(3,1,6,0), (-4,2,-1,10)\}$
Step 4: $\dim(\operatorname{Im}(T)) = 2$
Can someone please verify if this is correct?
Best Answer
You're right. A basis for the column space of the RREF would be $\{(1,0,0,0),(0,1,0,0)\}$. So the image has basis the corresponding columns of the original matrix.