2 3 Solutionb Let A be the matrix with v1v2v3 and v as colum

Solution

(b) Let A be the matrix with v₁,v₂,v₃ and v as columns. Then the RREF of A is

1

0

0

23

0

1

0

1/2

0

0

1

-3/2

Hence [v]_B = (23, 1/2, -3/2)^T

(c ) From part (b) above, we have v = (23)v₁+(1/2)v₂ –(3/2)v₃. Now, since T is a linear transformation, it preserves both vector addition and scalar multiplication. Hence T(v) =T((23)v₁+(1/2)v₂-(3/2)v₃) = (23)T(v₁)+(1/2) T(v₂ )- (3/2)T(v₃) = (23)(3v₁)+(1/2)(-2v₂ ) - (3/2)(5v₃) = (63)(1/3, 1/3, 1/3)-(2)(-1/2, 1/2,0) -5(3/2)(1/6,1/6 -2/6) = (6,6,6) +(1-1,0)+ (-5/2, -5/2, 5) = (9/2, 5/2,1).