3

Recall the following definitions (that you should have seen in first-year) concerning transformations.

A transformation T: R" →R" is a linear transformation if

1. Vu, v € R", T(u + v) = T(u) +T(v);

2. Vv € R" and Ve € R. T(cv) = cT(v).

Show the following theorem.

Let A E Mmn. Then the matrix transformation T₁ : R" →R" defined, for X € R", by

TA(X) = AX,

is a linear transformation.

[Hint: do not overthink this, this literally takes two short lines!]