Mathematics 1350H Linear algebra I: Matrix algebra
Trent University, Summer 2014
Solutions to Assignment #5
Determinants by way of Gauss-Jordan reduction
Given a square matrix A, we can compute a numb
Section 1.4 Applications 53
Using the law of sines, we have
f2 45°
nan _ lllel _ urn
\450 sin 45° sin 30° sin 105°
‘/ so
600 H H | H
r sin 45° 49(0.7071) r sin 30° 49(0.5)
r2f1+f2 |f1|=_—0
Prove that the given is a linear transformation using the definition (or the remark
http:/ltcconline.net/greenl/courses/203/Vectors/linearTransRn.htm
Give a counterexample to show that given is not a
Tutorial #11
MAT 188 Linear Algebra I Fall 2015
Solutions
Problems
(Please note these are from Holts Linear Algebra Text)
6.4 - # 7 Find the matrix A that has the eigenvalues 1 = 1, 2 = 0, and 3 = 1,
1. 2.2 Systems of Equations
Example 1.1. Solve the
x1
2x1
3x1
following system:
4x2
6x2
x2
3x3
5x3
4x3
Sol:
The augmented matrix of the system is
1 4 3 3
2 6 5 5
3 1 4 5
Row reducing
1 4
2 6
3 1
1. 2.3 Linear Independence and Spanning
4
2
1
Example 1.1. For which value of a are the vectors 2 ; 3 and 3 linearly
a
5
1
independent?
Sol:
By row reduction we get
1 2 4 | 0
1 2
4
| 0
2 3 3 | 0 R
1
Example 1.1. Let A = 2
3
invertible.
1. 3.3 Inverses
2 1
5 3. Find the values of a for which this matrix is
9 a
Sol:
Recall that a matrix is invertible if and only if it can be row reduced to In .
Example 0.1. A 3 5 matrix has rank 2. What is the nullity of the matrix?
Sol: We know that
rank(A) + nullity(A) = number of columns(A)
Thus
2 + nullity(A) = 5
Therefore, the nullity is 3.
Example 0.2.
DETERMINANTS
TERRY A. LORING
1. Determinants: a Row Operation By-Product
The determinant is best understood in terms of row operations, in my opinion. Most books
start by dening the determinant via fo
1. 1.3 Lines and Planes (II)
Question 1.1. Are the planes
x + 2y z = 3
and
2x 4y + 2z = 5
parallel or perpendicular?
Answer: Their normal directions are
2
1
n1 = 2 ; n2 = 4
2
1
Since n2 = 2n1 the
MATH 1350
Assignment 4
Solutions
1
Section 3.2
Exercise 6. We want
C1
1
0
0
0
+ C2
1
1
1
1
+ C3
0
0
This leads to a system of equations with
1 0 1
0 1 1
0 1 0
1 0 1
By row reduction we get
1
0
0
1
1
R
MATH 260 Homework 3 solutions
51. Let S be the linear space of innite sequences of real numbers x : px1 , x2 , . . .q. Dene the
linear map L : S S by
Lx : px1 x2 , x2 x3 , x3 x4 . . .q.
(a) Find a bas
A Geometric Review of Linear Algebra
The following is a compact review of the primary concepts of linear algebra. I assume the
reader is familiar with basic (i.e., high school) algebra and trigonometr
11. V6, [1/V3,1/\/3,1/\/2,0]
13. x/ﬁ 15. V6
17. (a) u - v is a scalar, not a vector.
(c) v - w is a scalar and u is a vector.
19. Acute 21. Acute 23. Acute
25. 60° 27. 388.100 29. zl4.34°
—4 l
—> —>
3
Mathematics 1350H Linear algebra I: Matrix algebra
Trent University, Summer 2013
Solutions to the Quizzes
Quiz #1. Wednesday, 15 May, 2013. [10 minutes]
1. Draw a sketch of the points (1, 0, 1) and (0
Quiz #7. Wednesday, 12 June, 2013. [12 minutes]
2
3
2
2 1 0
Let A = 4 1 1 0 1 5.
3
1 0 5
1. Use the Gauss-Jordan method to put A in row-reduced echelon form. [2]
2. Find a basis for two (2) of the fol
Mathematics 1350H Linear algebra I: Matrix algebra
Trent University, Summer 2014
Solutions to Assignment #3
Quadratic nonsense
1. Find a 2 2 matrix X =
Note: I2 =
1
0
0
1
a
c
b
d
with real entries suc
Mathematics 1350H Linear algebra I: Matrix algebra
Trent University, Summer 2014
Solutions to Assignment #4
Linear algebra for non-linear equations?
Recall that the general equation of a circle of rad
Mathematics 1350H Linear algebra I: Matrix algebra
Trent University, Summer 2014
Solutions to Assignment #2
Optimizing
Consider the region in R2 consisting of all the points that satisfy all of the fo
Mathematics 1350H Linear algebra I: Matrix algebra
Trent University, Summer 2014
Solutions to Assignment #1
Two classic puzzles
The two questions below are similar to problems posed in the Middle Ages
Mathematics 1350H Linear algebra I: Matrix algebra
Trent University, Summer 2014
[In Peterborough!]
Instructor
Department of Mathematics
Stefan Bilaniuk (pronounced Stefan B lan k)
oce: GCS 342
oce: G
Quiz #9. Wednesday, 19 June, 2013. [10 minutes]
A linear transformation T : R2 ! R2 is dened by
1
0
2
1
T
=
and T
=
.
2
1
1
0
3
1. Find T
. [5]
4
3
1
2
Solution. We will rst write
as a linear co
Quiz #3. Monday, 27 May, 2013. [15 minutes]
1. Find all the solutions, if any, to the following system of linear equations:
2x
x
3x
+
+
3y
y
y
+
+
z
z
z
=
=
=
6
1
1
[5]
Solution. Well go whole hog and
Mathematics 135H Linear algebra I: matrix algebra
Trent University, Fall 2007
Solutions to Quizzes
Quiz #1. Friday, 21 September, 2007. [5 minutes]
1. Find the acute angle between the vectors a = [2,
2
0
4 is in the span of 1 ,
Solution II. More systematically, note that, by denition,
6
1
1
1
0 , and 1 if there are scalars a, b, and c such that:
1
0
0
1
1
2
a1 + b0 + c1 = 4
1
1
0
6
Quiz #7. Friday, 16 November, 2007. [10 minutes]
1. Suppose A and B are invertible k k matrices. Solve the matrix equation
X1 A
1
= A B2 A
1
for the (invertible) k k matrix X. Simplify your answer as
Quiz #5. Wednesday, 5 June, 2013. [10 minutes]
1. Find the inverse, if any, of the following matrix:
2
2
4 1
2
3
1
3 5
1
3
2
1
[5]
Solution. As usual, we set up the super-augmented matrix and apply th