Linear Algebra Practice Problems Page 1 of 7

Linear Algebra Practice Problems

These problems cover Chapters 4, 5, 6, and 7 of Elementary Linear Alge-

bra, 6th ed, by Ron Larson and David Falvo (ISBN-13 = 978-0-618-78376-2,
ISBN-10 = 0-618-78376-8). Direct questions from Chapters 1–3 do not ap-
pear here, but the topics of Chapters 1–3 are very important because the
techniques covered there (Gaussian and Gauss-Jordan elimination, matrix
algebra, determinants) are essential for the later chapters.

1. True or False?
(a) A linear transformations is completely determined by its values on a basis for the domain.
(b) The kernel of a linear transformation is a subspace of the domain.
(c) The range of a linear transformation is a subspace of the co-domain.
(d) The rank of a linear transformation equals the dimension of its kernel.
(e) The nullity of a linear transformation equals the dimension of its range.
(f) A linear transformation T is one-to-one if and only if ker(T ) = {0}.
(g) If T : V → R5 is a linear transformation then T is onto if and only if rank(T ) = 5.
(h) If a linear transformation T : Rn → Rn is one-to-one, then it is onto and hence an isomorphism.
(i) If a linear transformation T : Rn → Rn is onto, then it is one-to-one and hence an isomorphism.

2. If T : R2 → R is a linear transformation from the plane to the real numbers and if T (1, 1) = 1 and
T (−1, 0) = 2, then T (3, 5) equals:
(A) −6 (B) −5 (C) 0 (D) 8 (E) 9

3. Let V = M2,3 be the vector space of all 2 × 3 matrices, and let W = M4,1 be the vector space of
all 4 × 1 column vectors. If T is a linear transformation from V onto W , what is the dimension of
{v ∈ V : T (v) = 0}?
(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

 
1 0 2
4. Let be the standard matrix of T . If e2 = (0, 1, 0) and v = (1, 1, 1), find
2 3 1
(a) T (e2 )
(b) T (v)
(c) T (x, y, z).

5. For each of the following linear transformations find its standard matrix.
(a) T : R2 → R3 , defined by T (x, y) = (3y, 2x, x − 4y).
(b) T : R2 → R2 is the reflection across x-axis.
(c) T : R2 → R2 is the orthogonal projection onto the line y = 2x.

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6. Consider the space C[−1, 1] of all continuous functions on [−1, 1] as an inner product space with the
inner product defined by Z 1
hf, gi := f (x)g(x) dx.
−1

Let φ and ψ be be given by:

φ(x) := 1 (−1 ≤ x ≤ 1), and ψ(x) := 3x − 1 (−1 ≤ x ≤ 1).

(a) Find the angle between φ and ψ.

(b) Find proj ψ φ (the orthogonal projection of φ onto ψ), and proj φ ψ (projection of ψ onto φ).
(c) Find an orthonormal basis for span({φ, ψ}).

7. Let T be the linear transformation defined by T (x) = Ax, where

 
1 2 2 3 1
A = 2 4 5 5 4 .
3 6 7 8 5

(a) Find a basis for ker(T ).

(b) Find nullity(T )
(c) Find a basis for range(T ).
(d) Find rank(T )

8. Let M2,3 be the vector space of all 2 × 3 matrices, let T : M2,3 → R3 be a linear transformation with
domain M2,3 . (Thus for each 2 × 3 matrix A, T (A) is a vector in R3 .) Suppose that the range of T
is the xz-plane {(x, y, z) : y = 0}. Find the rank of T and the dimension of ker(T ) (the nullity of T ),
giving detailed reasons for your answers.

9. Notation: If B is a given basis of a finite dimensional vector space V , then for every vector x in V ,
we write [x]B to denote the coordinates of x with respect to the basis B.
Now consider two bases B and B 0 for R2 , where B = {(2, 1), (3, 2)}, and B 0 is the standard basis.

(a) Suppose that the vector x ∈ R2 has coordinates (−2, 1) with respect to the (non-standard) basis B,
i.e. [x]B = (−2, 1). Find [x]B 0 , the coordinates of x with respect to the standard basis.
(b) Find the transition matrix from B to B 0 , i.e. find a matrix P such that P [x]B = [x]B 0 for every
vector x in R2 .
(c) Find the transition matrix from B 0 to B, i.e. find a matrix Q such that Q[x]B 0 = [x]B for every
vector x in R2 .
(d) Suppose that the vector x ∈ R2 has coordinates (−2, 1) with respect to the standard basis B 0 , i.e.
[x]B 0 = (−2, 1). Find [x]B , the coordinates of x with respect to the (nonstandard) basis B.
 
2 2 3 3
(e) If T : R → R has standard matrix A = , find the matrix of T relative to the basis B.
5 4

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10. For each matrix A, diagonalize A if possible, following the steps listed below.
       
2 2 0 2 0 0 2 1 0 2 1 0
(a) A = 0 2 0 (b) A = 0 2 1 (c) A = 0 2 1 (d) A = 0 3 1 .
0 0 1 0 0 1 0 0 2 0 0 4

(a) Find all eigenvalues and eigenvectors, and bases for each eigenspace.
(b) List algebraic and geometric multiplicities of each eigenvalue.
(c) Determine if A can be diagonalized.
(d) If A can be diagonalized, find a matrix P such that P −1 AP is diagonal.

11. For each matrix A given, orthogonally diagonalize A following the steps listed below.
     
2 −1 −1 3 1 0 0 1 1
(a) A = −1 2 −1  ; (b) A = 1 3 0  ; (c) 1 0 1 .
−1 −1 2 0 0 2 1 1 0

(a) Find the characteristic equation and all eigenvalues of A, listing the multiplicity of each eigenvalue.
(b) For each eigenvalue λ of A, find all the eigenvectors (the eigenspace) corresponding to it.
(c) Find a basis consisting of an orthogonal set of eigenvectors of A.
(d) Using the results above, orthogonally diagonalize A by finding a suitable orthogonal matrix P such
that P −1 AP is diagonal.

12. Consider the vectors u = (1, −1, −1, 0) and v = (1, −1, 0, −1) in R4 with the standard inner product
(i.e. the dot product). Let W be the subspace (of R4 ) spanned by u and v, and let W ⊥ be the
subspace of all vectors orthogonal to both u and v. Find an orthonormal basis w1 , w2 , w3 , w4 of R4
such that w1 , w2 is an orthonormal basis for W and w3 , w4 is an orthonormal basis for W ⊥ .

   
1 −1
13. In R3 , let S = span 0 ,  1.
1 0

(a) Find the orthogonal complement S ⊥ of the set S (all vectors orthogonal to all vectors of S).
(b) Find a linear transformation whose kernel is S and whose range is S ⊥ .
(c) Find a linear transformation whose kernel is S ⊥ and whose range is S.

14. Let T : R3 → R3 be the transformation on R3 which reflects every vector across the plane x+y+z = 0.
(a) List all eigenvalues of T .
(b) Describe all the eigenvectors of T .
(c) For each eigenvalue, find its algebraic and geometric multiplicity.
(d) Write down the standard 3 × 3 matrix of T .

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Multiple Choice Problems

15. The dimension of the subspace spanned by the real vectors

           
1 2 0 2 1 0
1 2 1 0 −2 0
 ,  ,  ,  ,  ,   is
0 0 0 0  0 0
0 0 0 3 8 0

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

16. The rank of the matrix  

1 2 3 4 5
 6 7 8 9 10
 
11 12 13 14 15 is
 
16 17 18 19 20
21 22 23 24 25
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

 
0 1 0
17. If M is the matrix 0 0 1, then M 100 is:
1 0 0
       
0 1 0 0 0 1 1 0 0 0 0 0
(A) 0 0 1 (B) 1 0 0 (C) 0 1 0 (D) 0 0 0 (E) none of the above
1 0 0 0 1 0 0 0 1 0 0 0

18. Let A and B be subspace of a vector space V . Which of the following must be subspaces of V ?
I. A + B := {x + y : x ∈ A and y ∈ B}
II. A∪B
III. A∩B
IV. {x ∈ V : x ∈
/ A}
(A) I and II only
(B) I and III only
(C) III and IV only
(D) I, II, and III only
(E) I, II, III, and IV

 
1 2
19. If A = then the set of all vectors x for which Ax = x is
0 −1
(A) {[a, b]T : a = 0 and b is arbitrary}
(B) {[a, b]T : a is arbitrary and b = 0}
(C) {[a, b]T : a = −b and b is arbitrary}
(D) {[0, 0]T }
(E) The empty set

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20. Let P3 be the vector space of all real polynomials that are of degree at most 3. Let W be the subspace
of all polynomials p(x) in P3 such that p(0) = p(1) = p(−1) = 0. Then dim(V ) + dim(W ) is
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

21. Let T be the transformation of the xy-plane that reflects each vector through the x-axis then doubles
the vector’s length. If A is the (standard) matrix of T , then A =
 
0 2
(A)
2 0
"√ #
2
1
(B) 2 √
1 − 22
"√ √ #
2 2
2 − √2
(C) √2 2
2 2
 
2 0
(D)
0 −2
 
0 −2
(E)
−2 0

22. Let V be the vector space of real valued functions defined on the real numbers and having derivatives
of all orders. If D is the mapping from V into V that maps every function in V to its derivative, what
are all the eigenvectors of D?
(A) All non-zero functions in V
(B) All non-zero constant functions in V
(C) All non-zero functions of the form keλx , where k and λ are real numbers
Pk
(D) All non-zero functions of the form i=0 ci xi , where k > 0 and the ci ’s are real numbers
(E) There are no eigenvectors of D

23. For what value (or values) of m is the vector (1, 2, m, 5) a linear combination of the vectors (0, 1, 1, 1),
(0, 0, 0, 1), and (1, 1, 2, 0)?
(A) For no value of m
(B) −1 only
(C) 1 only
(D) 3 only
(E) For infinitely many values of m

24. Which of the following sets of vectors is a basis for the subspace of Euclidean 4-space consisting of all
vectors that are orthogonal to both (0, 1, 1, 1) and (1, 1, 1, 0)?
(A) {(0, −1, 1, 0)}
(B) {(1, 0, 0, 0), (0, 0, 0, 1)}
(C) {(−2, 1, 1, −2), (0, 1, −1, 0)}
(D) {(1, −1, 0, 1), (−1, 1, 0, −1), (0, 1, −1, 0)}
(E) {(0, 0, 0, 0), (−1, 1, 0, −1), (0, 1, −1, 0)}

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25. Suppose that B is a basis for a vector space V of dimension greater than 1. Which of the following
statements could be true?
(A) The zero vector of V is an element of B.
(B) B has a proper subset that spans V .
(C) B is a proper subset of a linearly independent subset of V .
(D) There is a basis for V that contains no vector of B.
(E) One of the vectors in B is a linear combination of the other vectors in B.

         
0 1 3 0 6
26. If A is a 3 × 3 matrix such that A 1 = 0 and A 4 = 1, then the product A 7 is
2 0 5 0 8
       
0 −1 1 9
(A) 0 (B)  2 (C) −1 (D) 10 (E) Not uniquely determined
1 0 0 11

Proofs

Proofs must be correct, clear, complete, and precise.

27. Prove the following.

(a) Let T : V → V be a linear transformation and suppose that the set of vectors v such that T (v) = v
is a spanning set. Then T must be the identity mapping, i.e. T (v) = v for all v ∈ V .
(b) Any two eigenvectors (non-zero) having distinct eigenvalues must be linearly independent.
(c) Any number of eigenvectors (non-zero) with all distinct eigenvalues must be linearly independent.
(d) Let T : V → W be a linear transformation. Then:
• Ker(T ) is a subspace of V and Ran(T ) is a subspace of W .
• T is one-to-one if and only if Ker(T ) = {0} (i.e. if and only if dim(Ker(T )) = 0).
• If dim(W ) = n then T is onto if and only if Ran(T ) spans W if and only if dim(Ran(T )) = n.
• If T is one-to-one, then T maps linearly independent vectors to linearly independent vectors.
(e) Let T : V → V be a linear transformation. Then:
• If λ is an eigenvalue of T , then the eigenspace Eλ := {v | T v = λv} is a subspace of V .
• T is one-to-one if and only if 0 is not an eigenvalue of T .

28. Let u, v, w be vectors in a vector space V . Prove that:

(a) u and v are linearly independent if and only if u + v and u − v are linearly independent.
(b) The three vectors u − v, v − w, and w − u are linearly dependent.
(c) If u, v, w form a basis for V , then the vectors u, u + v, and u + v + w form another basis for V .

29. Show that if v1 , v2 , . . . , vn are non-zero vectors in an inner product space which are pairwise orthogonal
(i.e. vi ⊥ vj for 1 ≤ i 6= j ≤ n), then v1 , v2 , . . . , vn are linearly independent.

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30. Let V be the vector space of all real polynomials p(x). Let transformations T, S be defined on V by
T : p(x) → xp(x) and S : p(x) → p0 (x) = dxd
p(x). Interpret (ST )(p(x)) as S(T (p(x))), and similarly
for T S.
(a) Prove that S and T are linear transformations.
(b) Find the kernel and the range for each of S and T .
(c) Prove that ST is an isomorphism of V onto V (that is, ST is both one-to-one and onto).

The following equivalences are very useful. It may be instructive to practice the proofs for all the
equivalences below.

If T : Rn → Rn is a linear transformation with matrix A, then the following conditions are all equivalent:

1. T is one-to-one
2. The only solution of the homogeneous system Ax = 0 is the trivial solution x = 0
3. T has zero nullity, i.e. dim(Ker(T )) = 0

4. T has full rank, i.e. dim(Range(T )) = n

5. T is onto
6. The system Ax = b has a solution for every vector b
7. T is both one-to-one and onto

8. The system Ax = b has a unique solution for every vector b

9. T is an isomorphism of V onto V
10. T maps some basis to a basis

11. T maps every basis to a basis.

12. A is non-singular (or equivalently, A is invertible, i.e. A−1 exists)
13. The columns of A are linearly independent
14. The rows of A are linearly independent

15. The columns of A span Rn

16. The rows of A span Rn
17. det(A) 6= 0

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