. 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.
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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