Linear transformations and image

Question

I have some doubts on my linear algebra homework... The activity says:

Find the image of the following linear transformations without doing any calculations.

A) $T: \Bbb R^2 \to \Bbb R^2$ defined by $T((1;2))=(3;1)$ and $T((0;1))=(1;1).$

B) $T: \Bbb R^2 \to \Bbb R^2$ defined by $T((2;3))=(1;0)$ and $T((-1;2))=(0;1)$.

I know how to calculate the image of a linear transformation using the definition and some algebra, but the activity says "without doing any calculations", so I have to spot the image just "seeing" it. How can I do so?

Answer 1

You have a pair of vectors that you know are in the image. And since it is a linear transformation, all linear combinations of these two vectors are in the image.

Since these two vectors are linearly independent, they span $\mathbb R^2.$

Answer 2

Note that in this specific example $T$ is performed on vectors that span $\mathbb R^2$ which produce vectors that also span $\mathbb R^2$. Since the arrival space is also $\mathbb R^2$, the two transformed vectors for each example span their respective arrival spaces and thus are valid images.