Answer to Question #219572 in Linear Algebra for dan

Question

. let t:r^2->r^2 be a linear transformation for which ( 1,2)= (2,3 ) and ( 0,1)= (1,4 ). find a formula for t.

Answer 1

Solution.

$T(1,2)=(2,3); T(0,1)=(1,4).$

Let (a, b) ∈ R² . Since{${(1, 2),(0, 1)}$} is a basis of R² we determine c₁, c₂ such that

$(a, b) = c_1(1, 2) + c_2(0, 1)$.

That is

$a = c_1; b = 2c_1 + c_2.$

Solving this system, we see that $c_1 = a$ and $c_2 = b − 2c_1 = b − 2a.$

Therefore$(a, b) = a(1, 2) + (b − 2a)(0, 1).$

It follows that $T(a, b) = aT(1, 2) + (b − 2a)T(0, 1) = a(2, 3) + (b − 2a)(1,4) = (2a, 3a) + (b − 2a, 4b − 8a) = (b, 4b-5a).$

Answer: $T(a,b)=(b,4b-5a).$