Question

Let t : the set of real numbers2 → the set of real numbers2 be the linear transformation satisfying t(v1) = (4, 3), t(v2) = (−1, 1), where v1 = (1, 1) and v2 = (1, −1). find t(x1, x2) for an arbitrary vector (x1, x2) in the set of real numbers2. t(x1, x2) = correct: your answer is correct. what is t(8, −2)?

Answer 1

First let's see if (8, -2) can be written as a linear combination of (1, 1) and (1, -1): we want to find
`c_1,c_2` such that

`c_1(1,1)+c_2(1,-1)=(8,-2)\implies\begin{cases}c_1+c_2=8\\c_1-c_2=-2\end{cases}`

Easily done; we find
`c_1=3` and
`c_2=5`.

Since
`T` is linear, we have

`T(8,-2)=T(3(1,1)+5(1,-1))=3T(1,1)+5T(1,-1)=3(4,3)+5(-1,1)`

`T(8,-2)=(7,14)`