T R2x rightarrow M2 times 2 R is a linear transformation and

T: R^2[x] rightarrow M_2 times 2 (R) is a linear transformation and is given by T(a + bx + cx^2) = (a a + b a + b + c 3a + 2b + c). Find the matrix of T with respect to the standard bases S_R_2[x] = (1, x, x^2) S_M_2 times 2 (R) = ((1 0 0 0), (0 1 0 0), (0 0 1 0), (0 0 0 1)).

Solution

We have T(a+bx+cx²)=

a

a+b+c

a+b

3a+2b+c

Therefore, T(1) =

1

1

1

3

T(x) =

0

1

1

2

and T(x²) =

0

1

0

1

We know that the standard matrix of a linear transformation T: UV has columns which are images of the vectors inj the standard basis of U. Hence, here, the standard matrix of T: R₂(x) M_2x2(R) has T(1), T(x), T(x²) ( as above) as columns i.e. the the standard matrix of T is (T(1), T(x(),T(x²)) where T(1), T(x(),T(x²) are as above.

a	a+b+c
a+b	3a+2b+c