Download MATH 320 Practice Midterm 1 and more Exams Linear Algebra in PDF only on Docsity!
MATH 320 PRACTICE MIDTERM 1
This practice midterm is about twice the length that the actual midterm will be. You are encouraged to take it in exam conditions—no notes, calculator, book, etc., and keeping time constraints in mind. Solutions will be posted on Tuesday, Oct. 2. For all problems, you must fully explain your reasoning. Write in complete sentences when appropriate.
Problem 1 (20 points). Find all solutions of the system
x + 2y + 5z = 3 −x + z + 4w = − 1 − 3 x + 4z + 13w = − 3.
(Typo corrected on 10/03)
Problem 2 (20 points). Determine if the matrices ( 3 4 6 − 3 − 3 − 6
and
are row equivalent, i.e. if one matrix can be reduced to the other by elementary row operations.
Problem 3 (20 points). Find all values of x such that the vector
x
is a linear combination of the vectors
(^) , and
Problem 4 (20 points). Let P 3 denote the vector space of polynomials of degree at most 3 (you may assume that P 3 is a vector space). Prove that the set
{p(x) : p(1) = 3p(3)} ⊂ P 3
is a vector space. Either check all 10 axioms or explain why this is unnecessary.
Problem 5 (20 points). Determine a basis for the column space of the matrix
A =
What is the rank of A? What is the dimension of the space of solutions of the homogeneous system A~x = ~0?
1
Problem 6 (20 points). Consider the vectors
~v 1 =
(^) ~v 2 =
(^) ~v 3 =
(^) ~v 4 =
in R^3 , and put
S 1 = {~v 1 } S 2 = {~v 1 , ~v 2 } S 3 = {~v 1 , ~v 2 , ~v 3 } S 4 = {~v 1 , ~v 2 , ~v 3 , ~v 4 }. (1) Which sets Si are linearly independent? (2) Which sets Si span R^3? (3) Which sets Si are a basis for R^3?
Problem 7 (40 points). Consider
W =
a b c 0
: a − c = 0
⊂ Mat 2 × 2.
(Check for yourself that W is a subspace of Mat 2 × 2 .) (1) Determine a basis B for W , and compute dim(W ). (2) Consider the three matrices
A 1 =
A 2 =
A 3 =
in W. For each matrix Ai, compute RepB (Ai), where B is your basis from part (1). (3) Do the matrices {A 1 , A 2 , A 3 } span W? Are they linearly independent? Are they a basis?
Problem 8 (20 points). Consider a homogeneous system A~x = ~0, where
A =
a 11 ∙ ∙ ∙ a 1 n .. .
am 1 ∙ ∙ ∙ amn
Assume this system has a nonzero solution. (1) Prove or give a counterexample: the rows of A are linearly dependent. (2) Prove or give a counterexample: the columns of A are linearly dependent.
Problem 9 (20 points). Consider the following two subspaces of Mat 2 × 2 :
W 1 =
a 0 0 d
: a − d = 0
W 2 =
a b c d
: a + d = 0
(Again, check for yourself that these are subspaces). Show that Mat 2 × 2 = W 1 ⊕ W 2 is the direct sum of W 1 and W 2. (This problem should be easier after class on Monday.)
2
