Mathematics IA
Worked Examples
ALGEBRA: THE VECTOR SPACE Rn
Produced by the Maths Learning Centre,
The University of Adelaide.
May 1, 2013
The questions on this page have worked solutions and links to videos on
the following pages. Click on the link with

MATH 423
Linear Algebra II
Lecture 8:
Subspaces and linear transformations.
Basis and coordinates.
Matrix of a linear transformation.
Linear transformation
Denition. Given vector spaces V1 and V2 over a
eld F, a mapping L : V1 V2 is linear if
L(x + y) = L

Worksheet 7: Linear transformations and
matrix multiplication
14. Use the denition of a linear transformation to verify whether the
given transformation T is linear. If T is linear, nd the matrix A such that
T (x) = Ax for each vector x.
T (x1 ) = |x1 |;

Worksheet 14: Dimension and linear
transformations
1. Lay, 4.5.13.
Answer: The dimension of Col A is 3, the dimension of Nul A is 2.
2. Lay, 4.5.19.
Answers: (a) True (b) False (does not need to pass through the origin)
(c) False (the dimension is 5, as a

Subspaces
Sinan Ozdemir, Section 9
I did not get to make it to subspaces today in class, so I decided to make this study sheet for you guys to briey
discuss Sub Spaces.
1
Introduction
We all know what Vector Spaces are (ie. R, R2 , R3 , etc) and we also k

Worksheet 11: Subspaces
We will consider the following vector spaces:
Rn , the spaces we studied before;
Pn , the space of all polynomials in one variable of degree n;
P, the space of all polynomials.
14. Are the following sets subspaces of R2 ?
(1) Th

Harvard University, Math 20 Spring 2010, Instructor: Rehana Patel
1
Worksheet - Answers
Gaussian Elimination
February 5, 2010
Solve each of the following systems of linear equations, using the technique
of Gaussian elimination.
A.
3x + 2y + 3z 2w = 1
x+y+

Harvard University, Math 20 Spring 2010, Instructor: Rehana Patel
1
Gaussian Elimination
Worksheet
February 5, 2010
Solve each of the following systems of linear equations, using the technique
of Gaussian elimination.
A.
3x + 2y + 3z 2w = 1
x+y+z =3
x + 2

Harvard University, Math 20 Spring 2010, Instructor: Rehana Patel
1
Worksheet
Vector Spaces, Basis & Dimension
March 3, 2010
A. For each of the following sets of n-vectors, decide whether or not it is a real
vector space in Rn . If it is, nd a basis for i

Harvard University, Math 20 Spring 2010, Instructor: Rehana Patel
1
Worksheet - Answers
Linear Independence and Span
March 1, 2010
A. For each of the following sets of vectors in R3 , determine whether it is linearly
independent, and describe its span. If

Harvard University, Math 20 Spring 2010, Instructor: Rehana Patel
1
Worksheet - Answers
Inverses
February 26, 2010
A. Let A and B be invertible n n matrices. Use the denition of the inverse,
and any facts that you know about it, to show that
1. A
1
= A1 ,

Test 1 Review Solution
Math 342
(1) Determine whether cfw_(x, y, z) R3 : x + y + z = 1 is a subspace of R3 or not.
Solution: 0 + 0 + 0 = 1. Additive identity is not in the set so not a subspace.
(2) Is the dimension of Pm (F ) is m ? Why ?
Solution: Dimen

Mathematics 109, Linear Algebra
Winter 2006
Assignment 1 Solutions
1. Prove or disprove if the following sets are subspaces of R3 .
(a) The set S = cfw_(a, 0, 0)| a R is a subspace of R3 .
Proof: S is non-empty since 0 = (0, 0, 0) S.
The set S is closed u