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 boo
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 +
Assignment 1
[1pt] 1. What file formats are acceptable for assignments?
Assignments must be submitted as .pdf files, under 10MB in size.
[1pt] 2. Name one calculator approved by the Faculty of Mathema
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 cont
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
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,
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 individuall
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
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 counterexam
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 canonic
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.
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, w
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.
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
AMath 250, Fall 2016
Asst. 11 Solutions
Page 1 of 4
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AMath 250, F16
Assignment 10 Solutions
Page 1 of 4
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and make
sure that JavaScript is enabled. Yo
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 =
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 wer
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, b
Assignment 4
Due October 17, 2017 at 4pm
Covers Sections 4.6, 4.7, 7.1, plus instructor material
Recall: In class, we devoloped a more general theory of the matrix of a linear mapping. The results
wer
Assignment 6 Solutions
Due October 31, 2017 at 4pm
Covers Sections 7.4 and 8.1
[2pt] 1. Prove that hp, qi = p(1)q(1) + p(2)q(2) + p(3)q(3) defines an inner product on P2 .
First, we will show that hp,
Assignment 6
Due October 31, 2017 at 4pm
Covers Sections 7.4 and 8.1
[2pt] 1. Prove that hp, qi = p(1)q(1) + p(2)q(2) + p(3)q(3) defines an inner product on P2 .
[1pt] 2. Prove that h~x, ~y i = x1 y1
Assignment 5 Solutions
Due October 24, 2017 at 4pm
Covers Sections 7.2 and 7.3
[3pt] 1. For each of the following matrices, decide whether A is orthogonal by calculating AT A. If
A is not orthogonal,
Assignment 4 Solutions
Due October 17, 2017 at 4pm
Covers Sections 4.6, 4.7, 7.1, plus instructor material
Recall: In class, we devoloped a more general theory of the matrix of a linear mapping. The r
Assignment 5
Due October 24, 2017 at 4pm
Covers Sections 7.2 and 7.3
[3pt] 1. For each of the following matrices, decide whether A is orthogonal by calculating AT A. If
A is not orthogonal, indicate h
Assignment 6
1
[1pt] 1. Find a basis for the orthogonal complement of Span 2 .
5
[3pt] 2. Use the Gram-Schmidt Procedure to produce an orthogonal basis for
the subspace spanned by
1
1
1
1 5
2
Assignment 3
[2pt] 1. Find the coordinates of p(x) = 6 + x 3x2 with respect to the basis
B = cfw_1 + x2 , 1 x + 2x2 , 1 x + x2 of P2 .
[4pt] 2. Find the
changeofcoordinates
matrix
to and from the s
Assignment 2
[4pt] 1. Prove that the set
A = cfw_a0 + a1 x + a2 x2 + a3 x3 | 2a0 + a2 = 0, a1 + 4a3 = 0, a0 , a1 , a2 , a3 R
is a subspace of P3 .
[2pt] 2. Prove that the set B =
a b
c d
| a, b, c, d
Assignment 1
[1pt] 1. What file formats are acceptable for assignments?
[1pt] 2. Name one calculator approved by the Faculty of Mathematics for use
on the final exam.
[1pt] 3. What is the minimum scor
Assignment 7
[2pt] 1. Prove that hp, qi = p(1)q(1) + p(2)q(2) + p(3)q(3) defines an inner
product on P2 .
[1pt] 2. Prove that h~x, ~y i = x1 y1 x2 y2 does not define an inner product on R2 .
[4pt] 3.