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Linear Algebra – Mth 341
Archive – Fall 1992 Files
Feb 27, 2001
This archive contains the sample problems and tests from Mth 341 Fall 1992. The original test instructions, headers and formatting have not been preserved.
Contents
1 Sample Test 1
2 Test 1 2
3 Test 2 4
4 Draft Exam 5
5 Final Exam 7
6 Contact Information 9
1 Sample Test
Problem 1. Compute A^3 where
A =
Problem 2. Find the reduced row echelon canonical form of the matrix
A =
Problem 3. Use the Gauss-Jordan method to find all solutions of the system of linear equations
2 x 1 − 3 x 2 + x 3 − 4 x 4 + 2x 5 = 3 x 1 − 9 x 2 + 14x 3 − 17 x 4 + 10x 5 = 0 x 1 + x 2 − 4 x 3 + 3x 4 − 2 x 5 = 2.
Write your solution in vector form with as many parameters as required.
Problem 4. Compute the inverse of the matrix
B =
Problem 5. Let A =
and suppose A−^1 BA =
. Find B.
Problem 6. Find all values of t for which the following matrix is invertible
t 4 2 1 2 1 1 1 1
Problem 7. Give an example of a 2 × 2 nonzero matrix C such that C^2 =0.
Remark: Because of the capabilities of the calculators available today you should expect problems more like 5, 6 and 7 than 1 – 4, i.e., problems which require some thought on your part. This sample test has never been used as an actual quiz, so don’t use it to judge the length or content of my quizzes. It’s just a study aid, not a prognosticator.
2 Test 1
Problem 8. Find the 2 × 2 matrix A such that
A
and A
- the set of all real polynomials with degree at most 13
- the set of all real polynomials with degree at least 7
- the set of all real polynomials p(x) with p′(2) =
- the set of all real polynomials divisible by x^3 − 2 x^2 + 7x + 29
- the set of all continuous functions f on [0, 1] with
R 1
0 x
(^2) f (x) dx = 0
- the set of all n × n matrices [aij ] with
Pn
k=1 akk^ = 7
- the set of all points (x, y, z) in R^3 with x − y + z = 2
Just answer “yes” for the case of a vector space and “no” for the case where we do not have a vector space.
3 Test 2
Problem 16. Let A be the matrix
Find a basis for the null space of A (that is, a basis for ker(A)).
Problem 17. Let
b 1 =
5 , b 2 =
5 , b 3 =
t 1
For what values of t is B = {b 1 , b 2 , b 3 } a basis of R^3?
Problem 18. Let A be an m × n matrix. If AT^ A and AAT^ are both invertible prove that m = n and A
is invertible.
Problem 19. If
A =
then find (1) the rank of A and the (2) nullity (i.e., dim ker(A)).
Problem 20. Let {v 1 , v 2 , v 3 , v 4 } be linearly independent vectors in some vector space V and let
w 1 = 2v 1 + 4v 3 , w 2 = v 2 , w 3 = v 3 , w 4 = v 2 + v 3.
Suppose we apply the casting–out technique to the ordered set {w 1 , w 2 , w 3 , w 4 }. (1) Would any of the wk be discarded? (2) If any wk are discarded which would be the first one discarded? (3) What is the dimension of the span of {w 1 , w 2 , w 3 , w 4 }.
Problem 21. Let
B =
B′^ =
so B and B′^ are ordered bases of R^3. Find the change of coordinates matrix CB,B′ from B to B′.
Problem 22. Let V be the vector space of real polynomials of degree ≤ 3 and let W be the vector space
of real polynomials of degree ≤ 4. Define the linear operator T : V → W by T (p) = q if q(x) =
R x
0 p(t)^ dt. Let B = { 1 , x, x^2 , x^3 } and B′^ = { 1 , x, x^2 , x^3 , x^4 } so B is an ordered basis of V and B′^ is an ordered basis of W. Compute the matrix of T relative to these bases.
4 Draft Exam
This test is a draft exam. It may be too long or may emphasize the wrong material.
Problem 23. Consider the matrix
C =
The matrix C can be shown to have rank 3 (take my word for it ). Find a vector in R^4 which is not in the row space of C. Is there a vector in R^3 which is not in the column space of C? (Be sure to justify your answers.)
Problem 24. Compute the inverse of the matrix
B =
Problem 25. Consider the matrix
A =
Problem 31. Compute the determinant of the matrix
B =
1 0 0 s − 1 t − 2 2 0 0 3 1 1 2 − 1 2
Note your answer will be a function of s and t.
Problem 32. Find all eigenvalues and eigenvectors of the matrix
A =
Problem 33. Let A =
. Prove that A is not diagonizable.
5 Fi nal Exam
Problem 34. Consider the matrix
C =
The matrix C can be shown to have rank 3 (take my word for it ). Find a vector in R^4 which is not in the row space of C. Is there a vector in R^3 which is not in the column space of C? (Be sure to justify your answers.)
Problem 35. Compute the inverse of the matrix
B =
You must use row reduction. Show your answer using exact fractions, not decimals.
Problem 36. Consider the matrix
A =
Find the reduced row echelon Gauss–Jordan canonical form of the matrix A. What is the rank? What is the dimension of the row space? What is the dimension of the column space? Is the vector
in the row space? (Be sure to explain why or why not.)
Problem 37. Let P be the vector space of polynomials in the variable x. Let U ⊂ P be the subspace
consisting of all polynomials p(x) with degree at most 4 such that
Z 1
− 1
p(x) dx = 0,
Z 1
− 1
p(x)x dx = 0.
Then an ordered basis B for U is given by
p 1 (x) = 5x^3 − 3 x, p 2 (x) = 3x^2 − 1 , p 3 (x) = 5x^4 − 3 x^2.
Is the polynomial p(x) = 15x^4 + 10x^3 − 6 x^2 − 6 x − 1 in U? If it is then express it as a linear combination of the ordered basis B.
Problem 38. Let C be the vector space of all continuous real valued functions on the real line R. Let
W ⊂ C be the subspace spanned by
1 , sin(x), cos^2 (x), sin^2 (x), cos(2x).
(A) Use the method of casting out to extract an ordered basis for W from the given ordered spanning set. (B) What is the dimension of W?
Problem 39. Use the Gauss–Jordan method to find all solutions of the system of linear equations
3 x 1 − 2 x 2 + 2x 3 − 4 x 4 = 3 x 1 − 6 x 2 + 4x 3 − 7 x 4 = − 2 2 x 1 + x 2 − 4 x 3 + 2x 4 = 2
Write your solution in vector form with as many parameters as required.
Problem 40. Let P be the vector space of polynomials in x. Let W ⊂ P be the subspace consisting of
all polynomials with degree at most 6 and with at least a double root at 1 and at least a double root at −1. Find a basis for W.
Note: A γ is at least a double root of the polynomial p(x) if γ is a root of multiplicity at least two, i.e., if p(γ) = p′(γ) =0. Another way to express it is that γ is at least a double root of p(x) if and only if (x − γ)^2 is a factor of p(x).
Problem 41. Compute the determinant of the matrix
B =
1 0 0 s − 1 t − 2 2 0 0 3 1 1 2 − 1 2
