233 Problem Sets


Problem Set 1

1. Describe the intersection of the three planes u + v + w + z = 6 and u +w +z = 4 and u + w =2 (all in four-dimensional space).  Is it a line or a point or an empty set?  What is the intersection if the fourth plane u = -1 is included?  Find a fourth equation that leaves us with no solution. 

2. Under what conditions on y0, y1, y2 do the points (0,y0), (1,y1), (2,y2) lie on a straight line?

3. Below is a system of three linear equations in three unknowns with two symbolic constants, a and b.  By changing the values of a and b, we can get not only different solutions but also different numbers of solutions.  Use your calculator to reduce the augmented matrix of this system in order to answer the questions.
x + 2y+ az = 4, 2x - y + 3z = b, 3x - 4y + 2z = -3
For what values of a and b have
(a) no solutions?
(b) a unique solution?
(c) a line of solutions (i.e. one free variable)?
(d) a plane of solutions (i.e. two free variables)?

4. Which of these rules gives a correct definition of the rank of A?
(a) The number of nonzero rows in the row reduced matrix R.
(b) The number of columns minus the total number of rows in A.
(c) The number of columns minus the number of free columns in A.
(d) The number of 1s in R.

5. Use the information given to determine whether each linear system Ax=b is consistent.  If so, state the number of parameters in its general solution:
        Size of A        Rank(A)        Rank[A | b]
a)    3 x 3                3                    3
b)    3 x 3                2                    3
c)    3 x 3                1                    1
d)    5 x 9                2                    2
e)    5 x 9                2                    3
f)    4 x 4                0                    0
g)    6 x 2                2                    2

6. By trial and error find examples of 2 by 2 matrices such that
(a) A(A) = -I, A having only real entries
(b) B(B) = 0, although B ≠ 0
(c) CD = -DC, but CD ≠ 0
(d) EF = 0, although no entries of E or F are zero. 

7. A is 3 by 5, B is 5 by 3, C is 5 by 1, and D is 3 by 1.  All entries are 1.  Which of these matrix operations are allowed, and what are the results?  For those not allowed, explain why not. 
BA, AB, ABD, DBA, A(B+C)

8. Let L be a letter L matrix, which is a square matrix of 0s and 1s in which the 1s form the shape of the letter L.  For any square matrix A, experiment using your calculator with the product AL for A and L of various sizes, such as 4 by 4, 5 by 5, and 6 by 6, to help you answer the following questions.  Your answers must be valid for letter L matrices of every possible size.
(a) Describe the product AL, that is, what does L do to A when it is multiplied on the right of A?  Explain.
(b) What is L2?
(c) Find a general formula for Lp in terms of p and n, where L is n by n. 

Here are solutions to this first problem set. 

Problem Set 2

1. If every row of a 4 by 4 matrix contains the numbers 0, 1, 2, 3 in some order, can the matrix be symmetric?  Can it be invertible?  Explain.

2.  This matrix has a surprising inverse.  Find its inverse by row reduction on [A | I].  Repeat for a 5 by 5 "alternating matrix" and explain why this pattern will continue for any size "alternating matrix".
        [1 -1 1 -1]
A = [0  1 -1  1]
       [0  0  1  -1]
       [0  0  0   1]
3.  (a) What 3 by 3 matrix  will add row 3 to row 1?
     (b) What matrix adds row 1 to row 3 and at the same time adds row 3 to row 1? 
     (c) What matrix adds row 1 to row 3 and then adds row 3 to row 1?  Explain the differences. 

4. If a 4 by 4 matrix has det A = 1/2. find det(2A), det(-A), det(A2), and det(A-1) (if there are typesetting issues, the last two are square and inverse). 

5. We discussed a formula for 3 by 3 determinants with six terms, three going down and three going up subtracted.  There is a similar formula for 4 by 4 determinants, but it has 4! = 24 terms (not just eight).  Also the negatives are not where you may guess.  Count row exchanges to find these determinants.  Explain all. 
        [0  0  0  1]                                    [0  1  0  0]
        [0  0  1  0]                                    [0  0  1  0]
det   [0  1  0  0]   = +1      and     det    [0  0  0  1]  = -1
        [1  0  0  0]                                    [1  0  0  0]


                   [-1  4  -4]
6. Let A =  [  1 -3   1].  
                   [ 1 -2   0]

Compute A3 + 4A2 + 5A + 2I (feel free to use your calculator) to see that it equals 0.  Rewrite the equation A3 + 4A2 + 5A + 2I = 0 to find a formula for the inverse of A as a polynommial in A.  Caution:  Do not assume that the inverse exists, since that is part of what you must show.  Finally calculate the inverse of A using your formula (feel free again to use your calculator). 

                                                [a  b  c]
7. If the determinant of A =    [d  e  f] = k, find the determinant of these matrices (and explain your reasons):
                                                [g  h  i]

    [d  e  f]       [3a  3b  3c]      [a + g   b + h   c + i]         [  -3a      -3b     -3c  ]
a) [g  h  i]  b) [-d   -e   -f]   c)[   d          e         f   ]    d)  [   d           e        f   ]
    [a  b  c]      [4g  4h  4i]      [   g          h         i   ]          [g- 4d  h - 4e  i - 4f]

8. If the right side b is the last column of A, solve the 3 by 3 system Ax = b.  Explain how each determinant in Cramer's Rule leads to your solution x.

9. Consider the parallelogram PQRS where P = (2,1), Q = (4,2), R = (3,6), S = (1,5).  Use vector algebra throughout this problem.
(a)  Find the vertices of the translate of this parallelogram in which P is translated to the origin.
(b) Find the vertices of the translate of PQRS whose centre is at the origin.  (The "centre" is the common midpoint of its diagonals.)
(c) Find the vertices of the parallelogram that is obtained by rotating PQRS 180 degrees about the vertex P and making its sides twice as long.

Here are solutions to the second problem set (if someone wants to see a TeX file for this, please ask). 

Problem Set 3:  (there are 11 problems here; the problem set will be scored out of 40 total points)

1. The set of vectors {u1, u2, u3} is a basis for a subspace W of R4.  and {v1, v2, v3} is a basis for the same subspace.  A vector w in W  has the coordinates (2, 4, -3) relative to the first basis.  Find its coordinates relative to the second basis.  Feel free to use your calculator.
u1 = (1,-2,0,3), u2 = (0,-1,3,2), u3 = (2,-3,1,0), v1 = (1,-1,-3,1), v2 = (-1, 1,1,1), v3 = (1,-2,2,1).

2.  (a) Find a basis for the column space of the matrix A below.
     (b) Find a basis for the column space of matrix A by finding AT and then finding a basis for the row space.
     (c) Knowing that these two bases are for the same subspace, how must the two sets of basis vectors be related to one another algebraically?
        [1  1  -3  3]
        [-2 0   2  0]
A =  [-3 2 -1  3]
        [-2 1  0  1]
        [2  0 -2 -1]
Feel free to use your calculator to help. 

3. (a) Project b = (1,0,0) onto the lines through a1 = (-1,2,2), a2= (2,-1,2), and a3 = (2,2,-1).  Add the three projections p1+p2+p3. 
    (b) Project the vector d = (1,1) onto the lines through c1 = (1,0) and c2 = (1,2).  Add the projections p1+p2. 
    (c) Explain the difference between (a) and (b). 

4. If w1, w2, and w3 are independent vectors, show that the differences d1 = w2-w3, d2 = w1-w3, d3 = w1-w2 are dependent.  On the other hand, show that the sums s1 = w2 + w3, s2 = w1 + w3, and s3 = w1 + w2 are independent. 

5. Choose x = (x1, x2, x3, x4) in R4.  It has 24 rearrangements like (x2, x1, x3, x4) and (x4, x3, x1, x2).  Those 24 vectors, including x itself, span a subspace S.  Find specific vectors x so that the dimension of S is (a) 0, (b) 1, (c) 3, (d) 4. 

6.  Find a basis for each of these subspaces of R4:
(a) All vectors whose components are equal.
(b) All vectors whose components add to zero.
(c) All vectors that are perpendicular to (1,1,0,0) and (1,0,1,1).
(d) The column space and null space of U = [1 0 1 0 1]
                                                                       [0 1 0 1 0].

7.  True or false.  Please explain.
(a) If the columns of A are linearly independent, then Ax=b has exactly one solution for every b.
(b) A 5 by 7 matrix never has linearly independent columns.

8.  If we add an extra column b to a matrix A, then the column space gets larger unless __________.  Give an example in which the column space gets larger and an example in which it doesn't.  Why is Ax = b solvable exactly when the column space doesn't get larger by including b?

9.  Construct a matrix with the required property, or explain why you can't
(a) Column space contains (1,1,0), (0,0,1), row space contains (1,2) , (2,5).
(b) Column space has basis (1,2,3), nullspace has basis (3,2,1).
(c) Dimension of nullspace = 1 + dimension of nullspace of the transpose. 
(d) The nullspace of the transpose contains (1,3), row space contains (3,1).
(e) Row space = column space, nullspace ≠ nullspace of the transpose.

10. A is a p by q matrix of rank r.  Suppose there are vectors b for which Ax = b has no solution.
(a) What inequalities must be true between p, q, and r?
(b) How do you know that ATy = 0 has a nonzero solution?
 
11.  Explain why v = (1, 0 ,-1) cannot be a row of A and also be in the nullspace. 

Here are solutions to the third problem set. 

Problem Set 4

1. For the vector u = (7,7,7) and the plane W given by 2x + y + 3z = 0, use the projection of u onto W to find:
(a) The distance from the tip of u to the plane W
(b) a nonzero vector q orthogonal to every vector in W. 

2.  Consider the subspace of R5 spanned by u1 = (1, 0, 2, 0, -1), u2 = (0,1,1,-2,0) and u3 = (-1,2,0,1,-3).  Apply the Gram-Schmidt process to find an orthogonal basis of this subspace.  Now reorder them, use u3 first, then u1, then u2, applying the Gram-Schmidt process again.  Why does this yield a different orthogonal basis for the same subspace?  Feel free to use calculators for computations.

3. Find all vectors that are perpendicular to (1,4,4,1) and (2,9,8,2).

4. Why are these statements false?
(a) If V is orthogonal to W, then the orthogonal complement of V is orthogonal to the orthogonal complement of W.
(b) If V is orthogonal to W and W is orthogonal to Z, then V is orthogonal to Z.

5. Suppose S only contains the two vectors (1,5,1) and (2,2,2) (not a subspace).  Find a matrix so that the orthogonal complement to S is its nullspace.  Notice that the orthogonal complement to S is a subspace, despite S not being one.

6. If q1 and q2 are outputs from a Gram-Schmidt process, what were the possible input vectors a and b?

7. The cosine space F3 contains all combinations y(x) = A cos x + B cos 2x + C cos 3x.  Find a basis for the subspace that has y(0) = 0.

8. Find a basis for the space of polynomials p(x) of degree ≤ 3.   Find a basis for the subspace with p(1) = 0. 

9. Let W be the set of all upper-triangular 4 by 4 matrices.  Show that W is a subspace of M44.  Find a basis for W and determine its dimension. 

10.  Let V denote the set of all infinite sequences of real numbers (x1, x2, …, xn … ).  Define addition and scalar multiplication by components.  Show that V with these two operations is a vector space.  Be careful to verify all needed axioms.  This is tedious, but not difficult. 

Here are solutions to this fourth problem set. 


Problem Set 5

1. Find the 4 by 4 cyclic permutation matrix in which x = (x1, x2, x3, x4) is transformed to Ax = (x2, x3, x4, x1).  What is the effect of A squared?  Show that A cubed = the inverse of A. 

2. Find the 4 by 3 matrix A that represents a right shift:  (x1, x2, x3) is transformed to (0, x1, x2, x3).  Find also the left shift matrix B from R4 back to R3, transforming (x1, x2, x3, x4) to (x2, x3, x4).  What are the products AB and BA? 

3. What is the axis and the rotation angle for the transformation that takes (x1, x2, x3) to (x2, x3, x1)?

4. Which of these transformations is not linear for v = (v1, v2)?
(a) T(v) = (v2, v1)
(b) T(v) = (v1, v1)
(c) T(v) = (0, v1)
(d) T(v) = (0, 1)
Explain all. 

5. Which of these transformations satisfy T(v + w) = T(v) + T(w), and which satisfy T(cv) = cT(v)?
(a) T(v) = v/||v||
(b) T(v) = v1 + v2 + v3
(c) T(v) = (v1,2v2,3v3)
(d) T(v) = the largest component of v.

6. For these transformations of V = R2 to W = R2, find T(T(v))
(a) T(v) = -v
(b) T(v) = v + (1,1)
(c) T(v) = 90° rotation = (-v2, v1)
(d) T(v) = projection = 1/2(v1+v2, v1+ v2)

7.  Find the image and kernel of T
(a) T(v1,v2) = (v2, v1)
(b) T(v1, v2, v3) = (v1, v2)
(c) T(v1,v2) = (0,0)
(d) T(v1,v2) = (v1, v1)

8.  Suppose a linear T transforms (1,1) to (2,2) and (2,0) to (0,0).  Find a matrix for T. 

9.  Look over all of this problem set.  Which transformations are isomorphisms?  Justify your claim. 

Here are solutions to this fifth problem set. 

Problem Set 6

1. Use your calculator to find the eigenvalues and their geometric multiplicities for the n by n letter L matrix.  Explain why the eigenvalues have the geometric multiplicities claimed. 

2. Why are the eigenvalues of A equal to the eigenvalues of its transpose?  Do the eigenvectors need to be the same?

3. If B has eigenvalues 1, 2, 3,  C has eigenvalues 4, 5, 6, and D has eigenvalues 7, 8, 9, what are the eigenvalues of the 6 by 6 matrix
A = [B  C]
       [0   D] ?

4. Suppose A has eignevalues 0, 3, 5 with independent eigenvectors u, v, w.
(a) Give a basis for the nullpsace and a basis for the column space.
(b) Find a particular solution to Ax = v + w.  Find all solutions
(c) Show that Ax = u has no solution.  (hint:  if it had a solution, then something would be in the column space.)

5. Diagonalise the matrix A and find one of its square roots, a matrix such that its square is A.  How many square roots will there be? 
A = [5 4]
       [4 5]

6. If the eigenvalues of A are 1 and 0, write everything that must be true about the matrix A and its square.

7.  Suppose Ax = kx.  If k = 0, then x is in the nullspace.  If k ≠ 0, then x is in the column space.  Those spaces have dimensions (n - r) + r = n.  So, why doesn't every square matrix have n linearly independent eigenvectors?

8. M is any 2 by 2 matrix and A = [1 2]. 
                                                       [3 4]
The linear transformation T is defined by T(M) = AM.  What rules of matrix multiplication show that T is linear?

9.  Suppose T transposes every matrix M.  Try to find a matrix A that gives AM = T(M).  Show that no matrix A will do it. 

10.  Consider the vector space of 2 by 2 matrices.  What is a basis for this vector space?  What is its dimension?  Suppose T transposes every matrix M.  What is a matrix for this transformation with respect to this vector space?  Compare this to the previous question. 

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