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 number called the determinant of A,
usually denoted by |A|
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%—z35.87andllel =_—oz @2536
5111105 0.9659 51n105 0.965
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 linear transformation
http:/ltcconline.net/greenl/cours
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, and the corresponding
eigenvectors (1, 1, 0)T , (1, 2,
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 R2 2R1 ;R3 R1 0 1 5 | 0
1 5 a | 0
0 3 a4 | 0
1 2
4
| 0
1
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 .
1 2
2 5
3 9
1 2
R3 3R2
0 1
0 0
1
R2 2R1 ;R3 3R1
3
a
1
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. How small can the nullity of a 6x11 matrix can be?
Sol
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 formulas that are nearly impossible to use except on
very
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 normal vectors have the same direction. The planes are
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 basis for the nullspace of L. What is its dimension?
(b) W
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 trigonometry. The order of presentation is unconventional, with em
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
—> —>
31. Since AB - AC = 1 - 1 = 0, LBACis a
right angle. _1
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, 1, 1), and the line joining them. (A crude
sketch wil
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 such that X2 + 2X = 5I2 . [5]
is the 2 2 identity matrix.
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 radius r centred at the point (p, q) is
(x p)2 + (y q)2 =
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 following inequalities:
4 y 4
4 x 4
9 2x + y 9
9 2x y 9
9
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, and may well go
back farther than that.
1. Three men
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: GCS 337
hours: weekdays 09:00-12:30
Oce hours: Monday an
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 combination of
and
, and then use
4
2
1
3
the linearity
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 apply the Gauss-Jordan method, using matrix notation:
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, 1, 0] and b = 2, 1,
5 . [5]
Solution. Suppose is the ac
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
This boils down to checking if there is a solution to
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 much as possible. [5]
Solution. We work to isolate X. T
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 the Gauss-Jordan
algorithm:
2
2
4 1
2
2
R1 $ R2 1
42
=)
2
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 following three subspaces:
i. col(A)
ii. row(A)
iii. null(