Ma 125 WH-3 Due: March 4, 2024
Write solutions to the following questions, then take clear pictures or scans to upload on
Gradescope. Sufficient steps shown, organization of the solution, correct mathematical
notation and clearly stated reasoning where appropriate are all necessary.
1. Suppose T is a linear transformation such that
1 10 1 5 2 2
2 7 1 3 0 0
T 3 = −1 , T 1 = 1 , T −1 = 4
Find the matrix of T .
In other words, find the matrix A such that T (x) = Ax for each x in R3 .
2. Any cubic polynomial can be written as
1 a0 1
T
x a1 x
2 for scalars
x x2
p(x) = a0 + a1 x + a2 x2 + a3 x3 = a0 a1 a2 a3 3 = a2 3
x a3 x
a0 , a1 , a2 , a3 . Observe that differentiating p(x) corresponds to a linear map on the
coefficients, in other words
T
a0
1
a1
x
a2
p′ (x) = Tdiff a x2
3
for an appropriate linear map Tdiff.
(a) Find the matrix Adiff that represents Tdiff
(b) Similarly, integrating an arbitrary quadratic function
1
b0 1 b0
T T x
b1 x Z x b1
2 x2
q(x) = b0 + b1 x + b2 x = b2 x 7→
0 3
2
q(t)dt = Tint b2 x
corresponds to an appropriate linear map Tint. Find the matrix Aint that represents
Tint.
(c) Compute the products AintAdiff and Adiff Aint.
Optional: how can these products be interpreted in the context of ”differentiating then
integrating” vs. ”integrating then differentiating”?
Write solutions to the following questions, then take clear pictures or scans to upload on
Gradescope. Sufficient steps shown, organization of the solution, correct mathematical
notation and clearly stated reasoning where appropriate are all necessary.
1. Suppose T is a linear transformation such that
1 10 1 5 2 2
2 7 1 3 0 0
T 3 = −1 , T 1 = 1 , T −1 = 4
Find the matrix of T .
In other words, find the matrix A such that T (x) = Ax for each x in R3 .
2. Any cubic polynomial can be written as
1 a0 1
T
x a1 x
2 for scalars
x x2
p(x) = a0 + a1 x + a2 x2 + a3 x3 = a0 a1 a2 a3 3 = a2 3
x a3 x
a0 , a1 , a2 , a3 . Observe that differentiating p(x) corresponds to a linear map on the
coefficients, in other words
T
a0
1
a1
x
a2
p′ (x) = Tdiff a x2
3
for an appropriate linear map Tdiff.
(a) Find the matrix Adiff that represents Tdiff
(b) Similarly, integrating an arbitrary quadratic function
1
b0 1 b0
T T x
b1 x Z x b1
2 x2
q(x) = b0 + b1 x + b2 x = b2 x 7→
0 3
2
q(t)dt = Tint b2 x
corresponds to an appropriate linear map Tint. Find the matrix Aint that represents
Tint.
(c) Compute the products AintAdiff and Adiff Aint.
Optional: how can these products be interpreted in the context of ”differentiating then
integrating” vs. ”integrating then differentiating”?
