MATH 2940 Academic Excellence Workshop Week 13
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Circle each as either True or False.
(a) If A is a symmetric matrix, then A is invertible.
False
True or
(b) Any real symmetric matrix is diagonalizable.
o

MATH 2940 Academic Excellence Workshop Week 7
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Circle each as either True or False.
(a) The rank of a matrix is equal to the dimension of the row space.
(b) Let A be an m x n matrix. Then the dim(Row(A)

AEW Week 5 Solutions
(1) (a) The figure suggests that w = 2b1 b2 and x = 1.5b1 + .5b2, in which case, [w]B = and [x]B
= . To confirm [x]B = , compute 1.5b1 + .5b2 = 1.5 + .5 = = x
(b) If [x]B = , then x is formed from b1 and b2 using weights -1 and 3.
x =

MATH 2940 Academic Excellence Workshop Week 5
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. a) Let b1 = , b2 = , w = , x = , and B= cfw_b1, b2. Use the figure below to estimate [w]B and
[x]B. Confirm your estimate of [x]B by using it and cfw_b1, b2

Week 6 Solutions
(1) Suppose that are perpendicular nonzero vectors in We have the linear relationship ,
multiplying v to both sides gives us: Since by assumption then we must have c = 0. A similar
approach by multiplying w to both sides of the equation s

MATH 2940 Academic Excellence Workshop Week 4
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. C =
D=
a)
Compute CD.
b)
Compute DC.
c)
Suppose P is an invertible matrix and A = PBP-1. Solve for B in terms of A. (Hint:
Consider the associative property

MATH 2940 AEW Week 4 Solutions
(1)
(a) D = [ Compute:
C=, C=, C=
=,
=,
CD = C =
=
(b) Since D has 3 columns, and C has 2 rows, the product DC is undefined. It does not have
corresponding columns and rows and therefore cannot be multiplied.
(c) Left-multip

MATH 2940 Academic Excellence Workshop Week 2
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Decide whether each subset of is linearly independent or dependent
(a)
(b)
(c)
(d)
2. Suppose that and are vectors in .
(a) Show that if is a linearly indep

MATH 2940 Academic Excellence Workshop Week 3
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Write a matrix equation that determines the loop currents. (Feel free to solve the system
later if you finish the worksheet early)
2. Determine by inspectio

MATH 2940 Week 3 Solutions
1.
2. a) Compare the corresponding entries of the 2 vectors. The second seems to be -3/2 times the
first vector. However, this doesnt hold for the third pair of entries. Thus, neither of the vectors is
a multiple of the other, a

Math 2940 AEW Week 2 Solutions
(1) (a) There is no quick way to just look at the set of vectors and determine whether or not it is
linearly independent in this case. A sure-fire algorithmic approach is just to write the vector
equation as a matrix equatio

Math 2940 AEW Week 1 Solutions
(1) When the second equation is replaced by its sum with 3 times the first equation, the system
becomes
2x1 x2 = h
0 = k + 3h
If k + 3h is nonzero, the system has no solution. Therefore, the system is consistent for any
valu

MATH 2940 Academic Excellence Workshop Week 6
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. [Find the first half of the problem from another student] [Orthogonal vectors are of
great interest in chapter 6].
2. Find a basis for the vector space V sp

MATH 2940 Academic Excellence Workshop Week 6
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Prove that a set of two perpendicular nonzero vectors from is linearly independent when
[Find the second half of the problem from another student]
2. Find a

Week 11 Solutions
(1)
(a) Since A is triangular, its eigenvalues are 4 and 5.
For = 4: A-4I = , and reducing [A-4I 0] yields . The general solution is x2, and a basis for the eigenspace is
v1 = Since =5 must have only a one-dimensional eigenspace, we can

MATH 2940 Academic Excellence Workshop Week 12
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Let = and = . Compute and compare (a)-(d) without using the Pythagorean Theorem.
a) ,
b)
c)
d)
e) Show that if x is in both W and W, then x = 0.
2. a) Let

AEW Week 13 Solutions
(1) (a) False. Consider the zero matrix as a counterexample.
(b) True. The result is guaranteed by the spectral theorem (page 397 in your book).
(c) True. Although not obvious one can show that the sum of two symmetric matrices, and

MATH 2940 Academic Excellence Workshop Week 11
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. A = , B =
a) Diagonalize A.
b) Compute B8.
c) Construct a nondiagonal 2x2 matrix that is diagonalizable but not invertible.
2. Let B = cfw_b1, b2, b3 be a

Week 10 Solutions
(1) a) The standard matrix A of T is
A = Col A =
b) A = Nul A =
c) A = basis for Col A =
d) basis for Nul A =
e) rank A = 3, dim Nul A = 1 - check that rank A + dim Nul A = n
(2) a) Choose a column or row to expand across (here I chose 4

MATH 2940 Academic Excellence Workshop Week 8
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. C
2. A
3. (a)
4. A mouse is put in the maze below. Each time period the doors in the maze are opened and it is
allowed to move to another room. 50% of the t

MATH 2940 Academic Excellence Workshop Week 9
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Use Cramers Rule to solve the following system:
2. Use the adjugate to find the inverse of the following matrix
3. Show the following gives an equation of a

Week 8 AEW Solutions
(1) (a) P = , x0 =
(b) x4 =
(c) (P-I) = =
~
x2 = probability vector
Therefore 83.333% of days are sunny after a long period of time and 16.667% of days are rainy.
(2)
(a) FALSE The bottom row of P I is the negative of the sum of the o

MATH 2940 Academic Excellence Workshop Week 6
Parmeet Singh (ps732) & Alyse SanFilippo (as2578)
1. Prove that a set of two perpendicular nonzero vectors from is linearly independent when
[Orthogonal vectors are of great interest in chapter 6].
2. Find a b

MATH 2940 AEW Week 7 Solutions
(1) (a) TRUE. The rank of a matrix corresponds to the number of pivot positions in the echelon
form of the matrix. This is equivalent to the number of linearly independent columns or rows. So
rank of a matrix = number of piv

Week 9 Solutions
(1)
(2)
The adjugate is the transpose of the cofactor matrix so we get:
See: http:/en.wikipedia.org/wiki/Adjugate_matrix#3_.C3.97_3_generic_matrix
We can also calculate det(A) and we get
Hence,
.
(3) We do an algebraic check, evaluate the

Math 2940 AEW Week 1 Solutions
(1) When the second equation is replaced by its sum with 3 times the first equation, the system
becomes
2x1 x2 = h
0 = k + 3h
If k + 3h is nonzero, the system has no solution. Therefore, the system is consistent for any
valu