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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.
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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.)
(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
0 a^2 − 25 2 1 1 a^2 − 22 17 a + 3
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 ).
(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.
X =
1 + x + x^2 + x^3 , 1 − x + x^2 − x^3 , 1 + x^2 , −x − x^3