Question

2. Suppose that T is a linear transformation from R2 to R4
such that T ((1, 1)) = (3, −1, 4, −3) and T ((2, −1)) = (3, −2, −1, −3).
Determine the matrix of T (with respect to the standard bases).

3. Suppose that T is a linear transformation from P2 to P1 such that
T (x^2 + 1)
= x + 2, T (3x − 1) = x + 1, and T (x^2 + x + 1
) = x + 3.
Determine T (ax^2 + bx + c). (Here a, b, and c are any real numbers, so this essentially means
“figure out T.”)

4. For each of the following linear transformations, find a basis for the kernel and the range.

(a) T : P2 → R3, T (a + bx + cx^2)

= (a + b + c, b − c, a + 2b).

(b) T : R4 → R2, T ((a, b, c, d)) = (a, a − b + d).

5. For each of the following linear transformations, determine whether it is one-to-one, onto, both,
or neither.

(a) T : R2 → R3, T ((a, b)) = (2a, a + b, −b)

(b) T : P1 → P1, T (a + bx) = (2a − 2b) + (a + 3b)x.

6. Consider two linear transformations, T1 and T2, each from R2 to R2, determined by the following

T1 ((1, 0)) = (0, 2), T1 ((0, 1)) = (−1, −1)
and T2 ((1, 0)) = (1/2, 1), T2 ((0, 1)) = (2, −1)

(a) Show that the composition T2T1 is invertible.

(b) Determine the matrix of the inverse transformation of T2T1 (with respect to the standard basis).