Exam ID#000347-00
Course: MATH 115 Term: F2000 Type: M Solutions: S
Exam ID#000347-01
Course: MATH 115 Term: F2000 Type: M Solutions: S
Exam ID#000347-02
Course: MATH 115 Term: F2000 Type: M Solutions: S
Exam ID#000347-03
Course: MATH 115 Term: F2000 Type
Assignment 3
[3pt] 1. Let '1? = [ A: ] and 17 = I: 11 ]. Determine proj and perp.
Illustrate all four vectors on a graph.
. ,_,_ /a~rf._._ (3)(4)+(2)(1) 4 _ 10 4 _ 40/17
PmJu llgvt W l 1 J A E i 10/17
4_,- ' _ _ 3 40/17 _ 11/17
lilpui *prJq-ju
Lecture 4d
Calculating a Determinant
(pages 259-61)
The definition we have used for a determinant is known as expanding by cofactors. Even more specifically, we have expanded by cofactors along the first
row. But the first step in (potentially) making the
Lecture 4c
The Determinant of an n n Matrix
(pages 258-9)
Just as we calculated the determinant of a 3 3 matrix by looking at smaller
2 2 matrices within it, we calculate the determinant of a 4 4 matrix by
looking at 3 3 matrices within it. And in general
Lecture 4b
The Determinant of a 3 3 Matrix
(pages 256-8)
As I mentioned in the previous lecture, the determinant of a 2 2 matrix is
a value that determines whether or not the related system of equations has a
unique solution. It turns out that the notion
Lecture 4a
The Determinant of a 2 2 Matrix
(pages 255-6)
Suppose we were trying to solve a system of two linear equations in two unknowns:
a11 x1 + a12 x2 = b1
a21 x1 + a22 x2 = b2
We would row reduce the augmented matrix
a11
a21
a12
a22
b1
b2
Because we
Math 115 Fall 2014 Assignment + Quiz 10 Solutions
1
Assignment 10 solutions
("
1. Let R2+ =
a
b
)
#
a, b > 0, a, b R . We define addition and scalar multiplication on R2+ as follows:
"
a
b
"
#
c
d
#
"
=
ac
bd
#
"
,
t
a
b
#
"
=
at
bt
#
.
We claim that R2+
Math 115 Fall 2014 Assignment + Quiz 1 Solutions
1
Assignment 1 solutions
1. Consider the two vectors ~u =
2
3
, ~v =
2
1
.
(a) cfw_2 marks Determine a vector of length 13 that is in the same direction as ~u.
p
Solution. The length of ~u is 22 + (3)2 = 13
Math 115 Fall 2014 Assignment 11 Solutions
1. Calculate each of the following and express your answers in standard form.
(a) (1 3i)(2 + 7i)
Solution. (1 3i)(2 + 7i) = (1 3i)(2 7i) = 19 13i = 19 + 13i.
5i
(b)
2 + 3i
7
5 i 2 3i
7 17i
17
Solution.
=
= 2
i.
Lecture 4e
Elementary Row Operations and the Determinant
(pages 264-8)
We saw in the previous lecture that it is much easier to calculate the determinant
of a triangular matrix, or better yet, a matrix with a row or column of all zeros.
And weve already s
Lecture 4j
Matrix Inverse by Cofactors
(pages 274-6)
While Im sure you all think that computing the cofactors of every entry of
a matrix is fun, you must be wondering why we would need the cofactors of
every entry. After all, we only need to compute the c
Lecture 4i
The Cofactor Matrix
(pages 274-5)
As we continue our study of determinants, we will want to make use of the
following matrix:
Definition: Let A be an n n matrix. We define the cofactor matrix of A,
denoted cof A, by
(cof A)ij = Cij
That is, the
Assignment 4
[2pt] 1. Which of the following matrices are in row echelon form? For each
matrix not in row echelon form, explain why it is not.
1 1
4
2 7
(a) 0
0
0
3
4 5
4 8
0
0
2 1 2
1
0 1
8 3
1 0
0
4
1 4
(b) 0 0
0 0
1
(c) 0
0
(a) and (b) are in row ec
Assignment 8
Note: All the questions in this quiz will be related to the matrix A given below.
You should feel free to use results from one question when solving another. In
fact, its what I intend for you to do!
1
A = 5
2
[1pt] 1. Is ~x =
4
1
5
8
8
in
Assignment 12
2
Let A = 5
0
2
8
1 7
0
2
[3pt] (1) Find the eigenvalues of A.
The eigenvalues of A are the solutions to the equation det(A I) = 0. We
compute det(A I) as follows:
2
2
8
5 1
7
det(A I) = det
0
0 2
= (expanding along the third
row)
2
2
Assignment 2
x1
[2pt] 1. Show that the set A = x2 | x3 = (x1 x2 )2 is NOT a subspace
x3
of R3 .
2
2
10
Since 3 is an element of A (as 36 = (2 3)2 ), but 5 3 = 30
36
36
180
is not in A (as 180 6= (10 30)2 ), we see that A is not closed under scalar
multi
Assignment 10
3
[4pt] (1) Evaluate the determinant of 0
4
the row or column of your choice.
4 1
2
8 by expanding along
6
3
There are six possible correct answers, depending on which row or column you
choose. Ill give the solutions to doing the first row,
Lecture 4l
Area, Volume, and the Determinant
(pages 280-3)
The determinant has another interpretation completely separate from systems
of equations and matrices. It turns out that it can also be used to calculate the
area of a parallelogram (in R2 ), the
Lecture 4k
Cramers Rule
(pages 276-8)
While the cofactor method isnt the most practical way to compute the inverse
of a matrix, it does give us the useful formula that
A1 =
1
(cof A)T
det A
And we can now apply this formula to a system of equations. Becau
Lecture 1h
Spanning Sets
(pages 18-20)
A common way to define a subspace is through using a spanning set. But before
we get to this definition, we note the following.
Theorem 1.2.2 If cfw_~v1 , . . . , ~vk is a set of vectors in Rn and S is the set of al