Math 225
Assignment 1 Solutions
Question 1. (a) We need to nd all solutions of the vector equation
1
4
7
0
t1 2 + t2 5 + 8 = 0 ,
3
6
9
0
that is, all solutions of the homogeneous system
t1 + 4t2 + 7t3 = 0
2t1 + 5t2 + 8t3 = 0
3t1 + 6t2 + 9t3 = 0.
To do
Math 225
Assignment 2 Solutions
Question 1. (a) Let us call the set A. We claim that A is not a
subspace, since it is not closed under scalar multiplication. We need to
1
give a concrete counterexample. For example, x = 0 belongs to A,
0
but
1
2
1
x = 0
Math 225
Assignment 3 Solutions
Question 1. (a) Since dim P2 = 3, it suces to show that 1 + x2 , 1 +
x, x + x2 are linearly independent by a theorem from class (Theorem 4
(3) in Section 4.3 of the book). Suppose that t1 (1 + x2 ) + t2 (1 + x) +
t3 (x + x2
Math 225 Assignment 9
This assignment will not be marked
Question 1. For each of the following matrices, determine a diagonal matrix D
similar to the given matrix over C. Also determine a real canonical form B and
a change of coordinates matrix P such tha
Math 225 Midterm 1
Instructions:
No books, notes, or electronic devices are permitted. The only exception are Pink
Tie calculators, which are allowed, but not necessary to solve any of the questions.
You must justify all your answers.
You have 50 minut
Math 225 Midterm 1 Solutions
Question 1. (a) We need to nd t1 , t2 , t3 R such that
t1 (1 + x2 ) + t2 (1 + x x2 ) + t3 (x x2 ) = 1 + x + x2 .
Comparing coecients of powers of x, we
augmented matrix, which we row reduce:
1 1
0 1
0 1
1 1
1 1 1 1
obtain
Math 225 Midterm 2
Instructions:
No books, notes, or electronic devices are permitted. The only exception are Pink
Tie calculators, which are allowed, but not necessary to solve any of the questions.
You must justify all your answers.
You have 50 minut
Math 225
Assignment 8 Solutions
Question 1. (a) The general recipe is to multiply numerator and denominator by the complex conjugate of the denominator. We compte
2
2(3 + 5i)
1
3
5
=
= (6 + 10i) =
+ i.
(3 5i)
(3 5i)(3 + 5i)
34
17 17
Simarly, we obtain
13
PRACTICE FINAL SOLUTIONS
Math 225 Online Applied Linear Algebra 2
Instructor: Jennifer McKinnon
Exam Duration: 2.5 hours
Permitted Aids: Non-programmable calculator
INSTRUCTIONS TO CANDIDATES
1. This exams contains 11 questions over 14 pages (including th
PRACTICE FINAL
Math 225 Online Applied Linear Algebra 2
Instructor: Jennifer McKinnon
Exam Duration: 2.5 hours
Permitted Aids: Non-programmable calculator
INSTRUCTIONS TO CANDIDATES
1. This exams contains 11 questions over 15 pages (including this page,
a
Solution to Practice 1d
B2(a) We need to see if there are coecients t1 , t2 and t3 such that t1 (1 + x) +
t2 (x + x2 ) + t3 (1 x3 ) = 1. The left side becomes:
t1
+
t3
(t1 + t3 )
+
+
t1 x
t2 x + t2 x2
+
2
(t1 + t2 )x + t2 x
t3 x3
t3 x3
Setting the coecien
Lecture 1b
A Review of Matrices
In our attempts to solve spanning and linear independence problems in Rn , we
created a new object known as a matrix. We went on to study many properties
of matrices, but for now, lets review the properties of a matrix that
Lecture 1c
Addition and Scalar Multiplication of Polynomials
(pages 193-194)
Long before you were introduced to Rn and matrices, you were introduced to
polynomials. And somewhere along the way you were taught addition and scalar
multiplication of polynomi
Lecture 1d
Span and Linear Independence in Polynomials
(pages 194-196)
Just as we did with Rn and matrices, we can dene spanning sets and linear
independence of polynomials as well.
Denition: Let B = cfw_p1 (x), . . . , pk (x) be a set of polynomials of d
Lecture 1e
Vector Spaces
All of this leads us to notice that quite a lot of things that we frequently encounter have a common underlying structure. And so, instead of studying these
things individually, we instead will study them in general, based only on
Solution to Practice 1e
D1(b) To prove that the zero vector is unique, we assume that two vectors (say
a and b) both satisfy the dening property of the zero vector. That is, let a
and b be such that, for all x
x+a=x=a+x
x+b=x=b+x
and
Since x + b = x for a
Lecture 1a
A Review of Vectors in Rn
We began our studies of linear algebra by looking at collections of numbers.
Lets take a moment to review the denition and properties of Rn .
x1
.
Denition Rn is the set of all vectors of the form . , where xi R. Tha