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E2 Practice Problems

The document is an exam review covering various topics in linear algebra, including definitions of vector spaces, subspaces, and matrix properties. It contains practice problems that require students to provide technical definitions, determine the truth of statements, and give examples of mathematical objects. Additionally, it includes matrix-related questions involving determinants, null spaces, and linear transformations.

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mory yi
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17 views3 pages

E2 Practice Problems

The document is an exam review covering various topics in linear algebra, including definitions of vector spaces, subspaces, and matrix properties. It contains practice problems that require students to provide technical definitions, determine the truth of statements, and give examples of mathematical objects. Additionally, it includes matrix-related questions involving determinants, null spaces, and linear transformations.

Uploaded by

mory yi
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Exam 2 Review

Practice Problems
1. Complete the technical definition of each term.

(a) The column space of an m × n matrix A is. . .


(b) A set H contained in a vector space V is a subspace if. . .
(c) Let {v1 , v2 , . . . , vp } be a subset of a vector space V . The span of {v1 , v2 , . . . , vp }
is. . . [Hint: writing “Span{v1 , v2 , . . . , vp }” is insufficient]
(d) A set of vectors {v1 , v2 , . . . , vp } in an abstract vector space V is linearly dependent if. . .

2. Determine whether the following statements are True or False. Justify your response.

(a) Let ⃗b be a nonzero vector in Rm , and let A be an m × n matrix. Then the set of solutions
to A⃗x = ⃗b is a subspace of Rn .
(b) Let A be an n × n matrix. If there is a nonzero vector in the null space of A then the
columns of A do not span Rn .
(c) If A is a 3 × 5 matrix with three pivot columns, then Nul A = {0}.
(d) If A and B are n × n matrices and det(AB) = 0, then det(A) = 0 or det(B) = 0.
(e) The vector space R2 is a subspace of R3 .

3. Give an example of a mathematical object satisfying each description below. Justify your
response.

(a) A noninvertible, nonzero 3 × 3 matrix.


(b) A subset H of a vector space V which contains the zero element of V , but which does
not form a subspace of V .
(c) A set of elements in P2 which span P2 , but which are not linearly independent.
(d) A pair of matrices A and B such that AB = BA.

4. Let
 
1 0 −3 0 0
0 1 −2 0 −4
A= 
0 0 0 1 −1
0 0 0 a b

(a) Assume a = 0 and b = 1. Find a basis for Nul A.


(b) Find all a and b for which Nul A has a basis consisting of two vectors.

1
5. Let A be the following matrix, where aij is a real number for every i and j.

 
a11 a12 a13
A = a21 a22 a23 
a31 a32 a33

(a) Find a vector in Col A.


(b) Suppose A row reduces to
 
−a11 −a12 −a13
B =  2a11 + a21 2a12 + a22 2a13 + a23  .
−3a31 + a11 −3a32 + a12 −3a33 + a13

If det(B) = 12, find det(A).


(c) Suppose the reduced echelon form of A is
 
1 2 0
C = 0 0 1 .
0 0 0

Find a basis for Col A.

6. Let
  
a b
S= in M2×2 : a, b, and c are real numbers. .
−b c

(a) Does S contain the zero vector of M2×2 ? Explain.


(b) Is S closed under addition in M2×2 ? Explain.
(c) Does S form a subspace of M2×2 ? Explain.

 
2 −1
7. Let A = .
1 3

(a) Is A invertible? Explain.


(b) Let T : R2 → R2 be the linear transformation whose standard matrix is A. Is T one-to-
one? Is it onto? Explain.
 
5 0 0
8. Let A = −3 −7 0 .
8 5 −1

(a) Find det(A).


(b) Find A−1 .

2
9. Let A and B be 3 × 3 matrices, with det(A) = −3 and det(B) = 4. Find the following
determinants.

(a) det(B T )
(b) det(A3 )
(c) det(AB −1 A−1 )
(d) det(5AB)

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