MATH 304: Linear Algebra
Fall 2024
Texas A&M University
S. Cecchini
Final: More practice problems
1. Let
0 1
A=
1 0
(a) Find a nonsingular matrix X such that X −1 AX is diagonal.
(b) Use Part (a) to compute A6 .
(c) Use Part (a) to compute A−1 .
2. Let


1 0 0
A = −2 1 3 
1 1 −1
(a) Find a nonsingular matrix X such that X −1 AX is diagonal.
(b) Use Part (a) to compute A6 .
(c) Use Part (a) to compute A−1 .
3. Find eigenvalues and corresponding eigenspaces for each of the following matrices:
4 3
(1)
−3 4
4 5
(2)
2 1


3 1 2
0 1 −2
(3)
0 1 4


1 −2 2
2 0 2
(4)
3 −2 4
(5)
4. Let C([0, 1]) be the vector space of continuous functions on the interval [0, 1]. On
C([0, 1]) define the inner product
Z 1
⟨f, g⟩ =
f (x)g(x) dx.
0
Compute the angle between 1 + cos(πx) and sin(πx).
5. Let {u1 , u2 , u3 } be an orthonormal basis for an inner product space V and let
u = u1 + 2u2 + 2u3
v = u1 + 7u3 .
Determine the value of each of the following:
• ⟨u, v⟩,
• ∥u∥ and ∥v∥,
• the angle θ between u and v.
6. Let L : P2 → P2 be the linear map defined by
L (p) = p − p′ .
Determine the matrix representing L with respect to the basis {1, 1 + x, 1 + x + x2 }.