Math 3A – Final Practice Problems
Hengrong Du (University of California, Irvine)
Hengrong Du
University of California, Irvine
June 6, 2025
Math 3A – Final Practice Problems
June 6, 2025
1 / 12
Problem 9: Recurrence for Structured Tridiagonal Matrices
This question is about matrices with 3’s on the main diagonal, 2’s above,
and 1’s below:
3 2 0
0
3 2 0
1 3 2
3 2
0
A1 = [3], A2 =
, A3 = 1 3 2 , An =
0 1 3 · · ·
1 3
0 1 3
0 0 ··· ···
1
2
What are the determinants of A2 and A3 ?
Let Dn = det(An ). Use cofactors of row 1 and column 1 to find
constants a, b such that
Dn = aDn−1 + bDn−2
3
This recurrence is equivalent to:
Dn
a b Dn−1
=
Dn−1
1 0 Dn−2
Use the eigenvalues of this matrix to describe the growth of Dn . Also,
compute D5 .
Hengrong Du (University of California, Irvine)
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Problem 9 Workspace: Determinants and Recursion
(a) Compute det(A2 ), det(A3 ):
(b) Find constants a, b for recurrence:
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Problem 9 Workspace: Growth Rate and D5
(c) Growth of Dn from eigenvalues of
a b
1 0
Compute D5 :
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Problem 10: Diagonalization from Transformation Action
Let T : R3 → R3 be a linear transformation defined by:
7
4
16
T (⃗
e1 ) = 2 , T (⃗
e2 ) = 5 , T (⃗
e3 ) = 8 .
−2
−2
−5
1
2
Find a basis B for R3 consisting of eigenvectors of T such that the
matrix representation [T ]B is diagonal.
Find the diagonal matrix [T ]B .
Hengrong Du (University of California, Irvine)
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Problem 10 Workspace: Diagonalizing T
(a) Find a basis B of eigenvectors:
(b) Find the diagonal matrix [T ]B :
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Problem 11: Subspace and Orthogonal Complement
Consider the vectors
1
2
⃗v1 =
3 ,
4
1
2
⃗v2 =
−3 ,
4
1
2
⃗v3 =
3 ,
−4
3
6
⃗v4 =
3 .
4
1
Find a basis and the dimension of the space V = span(⃗v1 , ⃗v2 , ⃗v3 , ⃗v4 ).
2
Find the dimension of V ⊥ .
3
Find a basis for V ⊥ .
Hengrong Du (University of California, Irvine)
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Problem 11 Workspace: Parts (a)–(b)
(a) Basis and dimension of V = span(⃗v1 , ⃗v2 , ⃗v3 , ⃗v4 ):
(b) Dimension of V ⊥ :
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Problem 12: Orthogonality and Expansion in Basis
Let
0
⃗v1 = 3 ,
4
−5
⃗v2 = −4 ,
3
5
⃗v3 = −4
3
and let this set span R3 . Let
10
⃗x = −10 .
20
1
Check whether the set {⃗v1 , ⃗v2 , ⃗v3 } is orthogonal.
2
Write ⃗x = c1 ⃗v1 + c2 ⃗v2 + c3 ⃗v3 . Find the values of c1 , c2 , c3 .
Hengrong Du (University of California, Irvine)
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Problem 12 Workspace: Orthogonality and Coefficients
(a) Is the set {⃗v1 , ⃗v2 , ⃗v3 } orthogonal?
(b) Find coefficients c1 , c2 , c3 in ⃗x = c1 ⃗v1 + c2 ⃗v2 + c3 ⃗v3 :
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Problem 13: Projection Onto a Subspace
Let V ⊂ R3 be the subspace defined as the intersection of the planes:
x + 2y + 2z = 0,
4x − y − z = 0.
Find the orthogonal projection of
1
⃗v = 4
0
onto the subspace V .
Hint: You may find it easier to project ⃗v onto V ⊥ first.
Hengrong Du (University of California, Irvine)
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Problem 13 Workspace: Orthogonal Projection
Step 1: Find a basis for V ⊥ (normal vectors)
Step 2: Compute the projection of ⃗v onto V ⊥
Step 3: Subtract to find the projection onto V
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