MAT 175 Spring 2011: Quiz 11
Dr. Gaywalee Yamskulna
04/20/11
Name:.SOLUTIONS.
Instructions: Please read the questions carefully. You must write complete solutions to receive complete credit. (Due Friday.)
3
. Find the orthogonal projection of u onto the
4
Math 175
H. Jordon
Section 4.1: Subspaces
Denition: A set W of vectors from Rn is a subspace of Rn if
1. 0 W ,
2. if u, v W , then u + v W (we say that W is closed under vector addition), and
3. if u W and c is any scalar, then cu W (we say that W is clos
H. Jordon
Math 175
Matrix-Vector Products
Answer the following questions in groups of 3 to 4 people. Please use your textbook!
Denition:
Let A = [a1 a2 . . . an ] be a m n matrix, where ai denotes the ith column of A,
v1
v2
and let v = . be a n 1 vecto
H. Jordon
Math 175
Matrices and Vectors
Answer the following questions in groups of 3 to 4 people. Please use your textbook!
We will use capital letters for matrices and lower case letters for vectors.
1. What is a matrix? Give an example of a 2 3 matrix,
H. Jordon
Math 175
Matrix Multiplication
Answer the following questions in groups of 3 to 4 people. Please use your textbook!
Denition: Let A = [a1 a2 . . . an ] be a m n matrix, where ai denotes the ith column of A, and
let B = [b1 b2 . . . bp ] be a n p
Math 175
H. Jordon
Section 3.1: Calculating Determinants
Denitions: Let A be an n n matrix.
The notation Aij means the (n 1) (n 1) matrix obtained from A be deleting the ith row and
the j th column.
The (i, j )-cofactor cij = (1)i+j det Aij .
det A = a
Math 175
H. Jordon
Section 5.1: Eigenvalues and Eigenvectors
[
]
[]
]
3 2
1
2
. Determine Au and Av. Draw u, v, Au, Av in the
, and v =
1. Let A =
,u=
1
1
0
1
plane. What do you notice?
[
You might have noticed that Av = 2v. When this happens, that is, Av
Math 175
H. Jordon
Section 5.2: The Characteristic Polynomial
In this handout, we are going to nd the eigenvalues of given matrix A.
[
]
2
3
1. Let A =
.
3 6
(a) Let t be a variable. Find A tI .
(b) Determine det(A tI ).
(c) Solve det(A tI ) = 0.
(d) You
Math 175
H. Jordon
Section 5.3: Diagonalization
1
3
3
1. Let A = 3 5 3 .
3
3
1
(a) Determine the eigenvalues of A.
(b) Find a basis for each eigenspace.
(c) Form matrix P from the bases vectors you found in the previous question. Determine P 1 AP .
What d
Math 175
H. Jordon
Section 7.3: Basis and Dimension
Denition: Let V be a vector space. A nite set of vectors S = cfw_u1 , u2 , u3 , . . . , uk is linearly dependent if there exists scalars c1 , c2 , . . . , ck , not all zero, such that c1 u1 + c2 u2 + c3
Math 175
H. Jordon
Section 1.7
Denition: A set S of vectors S = cfw_u1 , u2 , u3 , . . . , uk is linearly dependent if there exists scalars
c1 , c2 , . . . , ck , not all zero, such that c1 u1 + c2 u2 + c3 u3 + + ck uk = 0.
A set S of vectors S = cfw_u1
Math 175
H. Jordon
Section 1.4
In Section 1.4, we use our row operationsmultiply by a scalar, interchange two rows, and add a multiple
of one row to anotherto nd the reduced row echelon form (rref) of a matrix. The process is called
Gaussian-Elimination.
MAT 175 Spring 2011: Quiz 1
Dr. Gaywalee Yamskulna
01/19/11
Name:.SOLUTION.
Instruction: Please read the questions carefully. You must write complete
solutions to receive complete credit.
1 1 5
1 1 1
1
and B =
. Find (3A 2 B )T . Please
371
25 4
show your
MAT 175 Spring 2011: Quiz 10
Dr. Gaywalee Yamskulna
04/6/11
Name:.SOLUTIONS.
Instructions: Please read the questions carefully. You must write complete solutions to receive complete credit. (Due Friday.)
1
0
1
1. (4 points) Let B = 1 1 2 and let T : R3
MAT 175 Spring 2011: Quiz 9
Dr. Gaywalee Yamskulna
03/30/11
Name:.SOLUTIONS.
Instruction: Please read the questions carefully. You must write complete
solutions to receive complete credit.
1. (4 points) Determine the dimension of the range and the dimensi
MAT 175 Spring 2011: Quiz 8
Dr. Gaywalee Yamskulna
03/23/11
Name:.SOLUTIONS.
Instruction: Please read the questions carefully. You must write complete
solutions to receive complete credit.
1. Let T : R4 R2 be a linear transformation dened by
x1
x
x1 3x2
MAT 175 Fall 2011: Quiz 7
Dr. Gaywalee Yamskulna
03/16/11
Name:.SOLUTIONS.
Instruction: Please read the questions carefully. You must write complete
solutions to receive complete credit.
1. (5 points) Show that the set
a
b
a2 + 2b2 1
is not a subspace of
MAT 175 Spring 2010: Quiz 2
Dr. Gaywalee Yamskulna
01/26/11
Name:.SOLUTION.
Instruction: Please read the questions carefully. You must write complete
solutions to receive complete credit.
1. (7 points) Determine the values of r and s for which the given s
Math 175
H. Jordon
Section 4.4: Coordinate Systems
In this section, we are going to learn how to represent vectors using dierent bases. If we write the vector
1
2 , it is understood that we mean the linear combination v = 1e1 +2e2 +3e3 ; that is, it is a