- #1

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## Homework Statement

Let T be the linear transformation from P2 (R) to P3 (R) defined by

[tex]T(f)=14\int_{0}^{x}f(t)dt + 7x.f'(x)[/tex]

for each

[tex]f(x)=ax^{2}+bx+c[/tex]

Determine a basis {g1, g2, g3} for Im(T).

## Homework Equations

as above

## The Attempt at a Solution

I evaluated the transformation function and simplified it, and found 3 linearly independent vectors for the basis, but I'm only asked for 1 and I think I've made a blunder somewhere.

[tex]\int_{0}^{x}f(t)dt=\frac{1}{3}ax^{3}+\frac{1}{2}bx^{2}+cx[/tex]

[tex]f'(x)=2ax+b[/tex]

[tex]T(f)=\frac{14}{3}ax^{3}+7bx^{2}+14cx+14ax^{2}+7bx[/tex]

[tex]T(f)=\frac{14}{3}ax^{3}+(14a+7b)x^{2} + (7b+14c)x[/tex]

[tex]T(f)= a(\frac{14}{3}, 14, 0)+b(0,7,7)+c(0,0,7)[/tex]

[tex]\therefore {(\frac{14}{3}, 14, 0),(0,7,7),(0,0,7)}[/tex] is a basis for Im(T)

I'd be pleased if someone could eyeball it for me and see if I haven't done something really silly!