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Linear Algebra Practice Exam #1, Exams of Linear Algebra

A practice exam for Linear Algebra course at Loyola University Chicago in Fall 2014. The exam consists of 5 problems, with 70% computational, 20% conceptual, and 10% proof questions. The exam covers topics such as row-reducing an augmented matrix, finding solution sets, determining the number of solutions for a system of equations, and identifying true/false statements related to linear algebra concepts. The exam also includes questions related to differential equations and polynomials.

Typology: Exams

2013/2014

Uploaded on 05/11/2023

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Linear Algebra Practice Exam #1
Loyola University Chicago Math 212.001 Fall 2014
There will be between approx. 5 problems on your exam. In terms of points, they will be
70% computational, 20% conceptual, and 10% proof. The proofs might be low- to medium-
difficulty. E.g., requiring you only to find a counter example to a statement, or do some
simple linear combinations to see the answer. (See your latest workshop for examples, and
#8 below for another.)
1. Solve by row-reducing an augmented matrix, displaying your solution set in multiple
forms (as all coordinates (w , x, y, z) of some form, and as a particular sum of vectors):
(a)
w+ 2x+ 3y+z= 1
5w6x9y+ 5z= 6
6w4x6y+ 6z= 7
(b)
a+b+c= 4
3ab+ 2c= 1
3a5b+c=2
2. For what values of adoes the system of equation with the following augmented matrix
have 0, 1 or infinitely many solutions?
1 3 5 2
0a225 2 1
1a222 17 a+ 3
.
3. Consider the differential equation(s):
f(x) = 2f(x) + 2xn1.(n)
(a) Find all polynomials f(x) in P2satisfying (2).
(b) Find all polynomials f(x) in P2satisfying (3).
4. Indicate if the following statements are True or False:
(a) It is never the case that X= [X].
(b) It is always the case that [[X]] = [X].
(c) If Xis linearly independent, then Xis a basis for [X].
(d) If XVfor some vector space V, and Xis linearly independent, then Xis a
basis for V.
(e) If XVfor some vector space V, and [X] = V, then Xis a basis for V.
(f) If XVfor some vector space Vof dimension n, and |X|=n, and Xis linearly
independent, then Xis a basis for V.
(g) If XVfor some vector space Vof dimension n, and |X|=n, and [X] = V,
then Xis a basis for V.
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Linear Algebra Practice Exam

Loyola University Chicago – Math 212.001 – Fall 2014

There will be between approx. 5 problems on your exam. In terms of points, they will be 70% computational, 20% conceptual, and 10% proof. The proofs might be low- to medium- difficulty. E.g., requiring you only to find a counter example to a statement, or do some simple linear combinations to see the answer. (See your latest workshop for examples, and #8 below for another.)

  1. Solve by row-reducing an augmented matrix, displaying your solution set in multiple forms (as all coordinates (w, x, y, z) of some form, and as a particular sum of vectors):

(a)

w + 2 x + 3 y + z = 1 5 w − 6 x − 9 y + 5 z = 6 6 w − 4 x − 6 y + 6 z = 7

(b)

a + b + c = 4 3 a − b + 2 c = 1 3 a − 5 b + c = − 2

  1. For what values of a does the system of equation with the following augmented matrix have 0, 1 or infinitely many solutions?  

0 a^2 − 25 2 1 1 a^2 − 22 17 a + 3

  1. Consider the differential equation(s):

f ′(x) = 2f (x) + 2xn^ − 1. (⋆n)

(a) Find all polynomials f (x) in P 2 satisfying (⋆ 2 ). (b) Find all polynomials f (x) in P 2 satisfying (⋆ 3 ).

  1. Indicate if the following statements are True or False:

(a) It is never the case that X = [X]. (b) It is always the case that [[X]] = [X]. (c) If X is linearly independent, then X is a basis for [X]. (d) If X ⊆ V for some vector space V , and X is linearly independent, then X is a basis for V. (e) If X ⊆ V for some vector space V , and [X] = V , then X is a basis for V. (f) If X ⊆ V for some vector space V of dimension n, and |X| = n, and X is linearly independent, then X is a basis for V. (g) If X ⊆ V for some vector space V of dimension n, and |X| = n, and [X] = V , then X is a basis for V.

(h) A system of equations with more equations than unknowns never has a solution. (i) A system of equations with more unknowns than equations will never have exactly one solution. (j) If A is an n × n matrix and the matrix equation A~x = ~b has a non-trivial solution, then the Gauss-Jordan form of A has n pivots. (k) If A is an invertible n × n matrix and ~b is a vector in Rn, then the matrix equation A~x = ~b has a unique solution ~x. (l) (5) True or False: If V is a vector space, every subspace of V must contain the vector ~0.

(m) (5) True or False: The set of

a b c

 (^) ∈ R^3 : a = b + 1

is not a subspace of R^3.

(n) It is possible to find 5 linearly independent vectors in R^3. (o) (Definition: The null space of a matrix is the set of vectors ~x solving the homo- geneous system A~x = ~0.) If A is a singular n × n matrix, then the null space of A is just {~ 0 }. (p) (Definition from class: a matrix is row/column deficient if it has fewer pivots than rows/columns in its Gauss-Jordan form.) If A is a row-deficient matrix, then the null space of A is never just {~ 0 }. (q) If A is a column-deficient matrix, then the null space of A is never just {~ 0 }. (r) The set of polynomials such that f (1) = f (2)^2 is a subspace of the vector space P of all polynomials.

  1. Show that the set of 2 × 2 matrices A such that the trace satisfies tr(A) = 0 is a subspace of M 2 × 2.
  2. Find a subset X′^ of the vectors X that forms a basis for [X]. If [X] 6 = P 3 , then extend X′^ to a basis for P 3.

X =

1 + x + x^2 + x^3 , 1 − x + x^2 − x^3 , 1 + x^2 , −x − x^3

  1. Find a tidy set Y of vectors in R^4 that forms a basis for [X]. If [X] 6 = R^4 , then extend Y to a basis for R^4.

X =

  1. (Definition: a polynomial ~p is a finite collection of monomials, e.g., ~p = a 0 + a 1 x + a 2 x^2 + · · · + arxr. The highest power of x occurring in ~p with nonzero coefficient is called the degree of ~p.) Show that the vector space P of all polynomials is infinite dimensional. (Hint: argue by contradiction, starting with a finite basis for P.)