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
5w−6x−9y+ 5z= 6
6w−4x−6y+ 6z= 7
(b)
a+b+c= 4
3a−b+ 2c= 1
3a−5b+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
0a2−25 2 1
1a2−22 17 a+ 3
.
3. Consider the differential equation(s):
f′(x) = 2f(x) + 2xn−1.(⋆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 X⊆Vfor some vector space V, and Xis linearly independent, then Xis a
basis for V.
(e) If X⊆Vfor some vector space V, and [X] = V, then Xis a basis for V.
(f) If X⊆Vfor some vector space Vof dimension n, and |X|=n, and Xis linearly
independent, then Xis a basis for V.
(g) If X⊆Vfor some vector space Vof dimension n, and |X|=n, and [X] = V,
then Xis a basis for V.
