Answer to Question #219572 in Linear Algebra for dan

Question #219572

. 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.


1
Expert's answer
2021-09-07T12:17:25-0400

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;\n\n b = 2c_1 + c_2."

Solving this system, we see that "c_1\t= a" and "c_2 = b \u2212 2c_1 = b \u2212 2a."

Therefore"(a, b) = a(1, 2) + (b \u2212 2a)(0, 1)."

It follows that "T(a, b) = aT(1, 2) + (b \u2212 2a)T(0, 1) = a(2, 3) + (b \u2212 2a)(1,4) = (2a, 3a) + (b \u2212 2a, 4b \u2212 8a) = (b, 4b-5a)."

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


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