Question

Let T: R2→ R2 be a linear transformation that sends the vector u=(5,2) into (2,1) and maps v=(1,3) into (−1,3). Use properties of a linear transformation to calculate

T(−5u)=( ..... , ..... ), T(8v)=( ..... , ..... ), T(−5u+8v)=( ..... , ..... )

Answer 1

`T(5u) =(10,5)`

`T(8v) =(8,24)`

`T(5u+8v) = (18,29)`

Explanation:

Since it is not clear to which vector is v mapped, we will asumme that T(v) = (1,3).

Recall that a linear transformation has the following properties. For vector a,b and scalar
`\lambda`, we have that

`T(a+b) = T(a) + T(b), T(\lambda a ) = \lambda T(a)`.

Since we have that u=(5,2) v=(1,3) and T(u) = (2,1) T(v) = (1,3). Then

`T(5u) = 5T(u) = 5(2,1) = (10,5)`

`T(8v) = 8T(v) = 8(1,3) = (8,24)`

`T(5u+8v) = T(5u)+T(8v) =(10,5)+(8,24) = (18,29)`