University of Ontario Institute of Technology (UOIT)
ALGEBRA Math1850

Fall 2015
MATH 1850U Midterm 2 November 2014 Page 2 of 8
1. Consider the vector u = (—4, —12, 8) and the set S : {V1,v2, V3} where
v1:(1,1,1), V2 2 (5, —3,1), V3 : (2,1,2).
5 is a basis fer 1R3 (this, you DO NOT have to check).
(a) [7 marks] Write the vector u as
University of Ontario Institute of Technology (UOIT)
ALGEBRA Math1850

Fall 2015
MATH1850U: Chapter 1 cont.
1
LINEAR SYSTEMS cont
Gaussian Elimination (1.2; pg. 11) cont.
Recall: Last day, we introduced Gaussian and GaussJordan Elimination for rowreducing a matrix. Lets get some more practice at this.
More Examples: Lets do some more
University of Ontario Institute of Technology (UOIT)
ALGEBRA Math1850

Fall 2015
MATH1850U: Chapter 1 cont
1
LINEAR SYSTEMS cont
Introduction to Systems of Linear Equations (1.1; pg.2) cont
Recall: Last day, we introduced the elementary row operations that help us replace a
system of equations with another system that has the same sol
University of Ontario Institute of Technology (UOIT)
ALGEBRA Math1850

Fall 2015
MATHISSOU: Chapter 4 cont. 1
GENERAL VECTOR SPACES cont.
S ubsgaces (Section 4.22
Deﬁnition: A subset Wof Vis called a subspace of Vif W is itselfa vector space under
the addition and scalar multiplication deﬁned on V.
\‘F CLAY; me in b.) we» an? =‘G+J. '
University of Ontario Institute of Technology (UOIT)
ALGEBRA Math1850

Fall 2015
MATH1850U: Chapter 6 cont.
1
INNER PRODUCT SPACES cont.
GramSchmidt Process; QRDecomposition (Section 6.3)
Definition: A set of vectors in an inner product space is called an orthogonal set if all
pairs of distinct vectors in the set are orthogonal. An
University of Ontario Institute of Technology (UOIT)
ALGEBRA Math1850

Fall 2015
MATH l 850le Chapter 5 l
EIGENVALUES AND EIGENVECTORS
Eigenvalues and Eigenvectors (Section 5.1)
Deﬁnition: IfA is an ax a matrix, then a nonzero vector x in R” is called an eigenvector
ofA if Ax is a scalar multiple of x; that is
Ax = ﬁx
for some scalar
University of Ontario Institute of Technology (UOIT)
ALGEBRA Math1850

Fall 2015
Formulas that will be provided during midterm #2, if needed:
u v u v cos( )
vector component of u along a:
proja u
u a
a
2
a
vector component of u orthogonal to a: u  proja u u
Wronskian: W ( x )
f1 ( x )
f1( x )
f 2 ( x)
f 2( x )
f1( n 1) ( x)
f 2( n