FINAL EXAMINATION SOLUTION
MATA23H3 - Linear Algebra I
1. (a) [5 points] If a, b, c R3 show that
a (b c) =
a
b
c
Solution: Let a = [a1 , a2 , a3 ], b = [b1 , b2 , b3 ], c = [c1 , c2 , c3 ]
LHS
= a (b c) = [a1 , a2 , a3 ]
= a1
=
b2 b3
c2 c3
a2
a1 a
University of Toronto Scarborough
Department of Computer & Mathematical Sciences
Midterm Test
MATA23H3 Linear Algebra I
Examiners: E. Moore
K. Smith
Date: March 5, 2014
Duration: 100 minutes
1. [12 points]
(a) Find the length of the vector joining the poi
* Sorry - Solutions Are Not Provided *
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
Final Examination
MATA23 - Linear Algebra I
Examiner: R. Grinnell
Date: August 21, 2012
Time: 7:00 pm
Duration: 180 minutes
Provid
University of Toronto at Scarborough Department of Computer & Mathematical Sciences
Midterm Test MATA23 Linear Algebra I Examiner: N. Cheredeko Date: February 11, 2005 Duration: 110 minutes
1. [7 points] (a) Give the denition of the span of n vectors. (b)
Euclidean nspace, Rn, is dened as
Rn = cfw_ (x1 , x2, , xn) | xi R, i = 1, 2, , xn
set of all ntuples
Each ntuple of real numbers (Rn) can be viewed as a point (x1, , xn)
or a vector [x1, , xn]. The ith entry, xi, of the vector is called the
ith componen
University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MATA23H3 Linear Algebra I Summer 2014
Assignment #8 Solutions to Selected Problems
4.1 # 4
21 4
= (21)(7) (10)(4) = 147 + 40 = 187
10 7
4.1 # 5 The vector a = b c has compo
University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MATA23H3 Linear Algebra I Summer 2014
Assignment #9 Solutions to Selected Problems
4.2 # 4 We compute by cofactor expansion along column 3:
4 1 2
3 1
4 1
3
1 0 =2
+
= 2(7)
University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MATA23H3 Linear Algebra I Summer 2014
Assignment #10 Solutions to Selected Problems
4.3 # 4 We use column operations to simplify:
3 2 1 1
3 5 1
7
3 2 1
0
0
1 0
0
3
1 6
= 2
University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MATA23H3 Linear Algebra I Summer 2014
Assignment #7 Solutions to Selected Problems
2.3 # 10 Note that
1
([1, 1] [1, 1])
2
1
[0, 1] = ([1, 1] + [1, 1])
2
[1, 0] =
Hence
T ([
University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MATA23H3 Linear Algebra I Summer 2014
Assignment #6 Solutions to Selected Problems
2.1 # 12
2
3
1
1
0
0
To nd a basis of the
5 1 6
1 1
2 7 2 2 5
1 2 0
3 2
1
2
0
1
0
1
University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MATA23H3 Linear Algebra I Summer 2014
Assignment #5 Solutions to Selected Problems
1.6 # 2 The zero vector [0, 0] is not contained in the set cfw_[x, x + 1] | x R, hence it
University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MATA23H3 Linear Algebra I Summer 2014
Assignment #4 Solutions to Selected Problems
1.4 # 24 The augmented matrix of the system is
1 2 3
1 2
3 6 8 2 1
1 2 3
1
2
0 0
1 5 5
1
University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MATA23H3 Linear Algebra I Summer 2014
Assignment #3 Solutions to Selected Problems
1.3 # 4 B + C is not dened since B is 2 3 and C is 3 2, and matrix addition is only
dened
University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MATA23H3 Linear Algebra I Summer 2014
Assignment #2 Solutions to Selected Problems
1.2 # 4
v 2u = [2, 1, 1] 2[1, 3, 4]
= [4, 5, 9]
(4)2 + (5)2 + (9)2
= 122
=
1.2 # 6
4
4
w
University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MATA23H3 Linear Algebra I Summer 2014
Assignment #1 Partial Solutions
1.1 # 4
v = 2i j + 3k
w = 3i + 5j + 4k
v + w = (2i j + 3k) + (3i + 5j + 4k)
= (2 + 3)i + (1 + 5)j + (3
Eigenvalues and Eigenvectors
Example: Let T : R2 R2 be the reflection about the line y = 2x.
1
c
2011
by Sophie Chrysostomou
DEFINITION: Let A be an n n matrix. A scalar is an eigenvalue of A if there is a
nonzero vector v Rn , such that Av = v. In this c
University of Toronto Scarborough
Department of Computer and Mathematical Sciences
MATA23H3 Linear Algebra I
Summer 2014
Course Information
Instructor:
J. Fisher
[email protected]
Oce: IC467
Homepage:
All announcements, assignments, etc. will be pos
Denition: Let A Mn(R) and let be an eigenvalue of A. The
solution set of the linear system (A I) x = 0 is called the
eigenspace associated to the eigenvalue and is denoted E .
Theorem: Let A Mn(R).
(i) If is an eigenvalue of A with v as a corresponding ei
Theorem: Let A be a n k matrix. The linear system A x = b is
consistent if and only if the vector b ( Rn) is in the span of the
column vectors of A.
Theorem:
(Solutions of A x = b)
Let A x = b be a linear system, and let A b B c , where
B is in row-echelo
Let a = [a1, a2, a3], b = [b1, b2, b3] R3 . Then the vector
a1 a2
a1 a3
a2 a3
e1
e2 +
b1 b3
b2 b3
e3
b1 b2
is perpendicular to a and b. It is represented by the symbolic matrix
e1
e2
e3
a1
a2
a3 = a b ,
b1
b2
b3
and called the cross-product of a and b.
P
Denition: Let A Mn(R) and let be an eigenvalue of A. The
solution set of the linear system (A I) x = 0 is called the
eigenspace associated to the eigenvalue and is denoted E .
Theorem: Let A Mn(R).
(i) If is an eigenvalue of A with v as a corresponding ei
Denition: Let v 1, v 2, , v k be vectors in Rn. The span of
these vectors is the set of all linear combinations of these vectors and
is denoted by sp(v 1, v 2, , v k ).
sp(v 1, v 2, , v k ) = cfw_1 v 1 + 2 v 2 + + k v k | 1, , k R.
Denition: Let v = [v1,
Denition: (matrix addition)
Let A = aij , B = bij
Mn,k (R). The matrix sum, A + B,
is the n k matrix C = cij
where cij = aij + bij .
Denition: The n k matrix whose entries are all zero, is called
the zero matrix and is denoted by O.
Denition: (scalar mul
Denition: A function T : Rk Rn which satises
1.
T (u + w) = T (u) + T (w), for all u, w Rk .
(preservation of addition)
2.
T ( u) = T (u), for all u Rk , R.
(preservation of scalar multiplication)
is called a linear transformation.
Theorem: Let T : Rk Rn
Denition: The linear system A x = 0 is said to be homogeneous.
The solution x = 0 is called the trivial solution.
Nonzero solutions are called nontrivial solutions.
Theorem: If v 1, v 2, , v k are solutions of the homogeneous linear
k
system A x = 0, the
w2
w1
w3
w2
w3
2w1
e3
w2
w3
w1
Denition: Let cfw_w1, w2, , wk be a set of vectors in Rn. A
dependence relation in this set is an equation of the form
1 w 1 + 2 w 2 + + k w k = 0
with at least one i = 0.
If such a dependence relation exists, then cfw_w1,
The non-invertible linear transformations T : R2 R2 are usually
called collapsing linear transformations.
They collapse R2 into a line or a point.
We say that v is projected onto p on a line through the origin if
(v p) p. The corresponding matrix is calle
Cramers Rule: Let A M(R) be
invertible. The linear system
n
b1
x1
b2
x2
A x = b where x = , b = has a unique solution
.
.
.
.
.
.
bn
xn
given, in components, by
xk =
det Bk
, k = 1, 2, , n
det A
where Bk is the matr
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A23
Winter 2017
MATA23H COURSE INFORMATION
Welcome to MATA23! In this course we will study systems of linear equations, matrices, Gaussian elimination; basis, dimensio
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
Solution of Term Test - MATA23H (Linear Algebra I)
Examiners: Sophie Chrysostomou
Xiamei Jiang
Date: Friday, March 3, 2017
Duration: 110 minutes
1. [15 points] Show all yo
Final Examination
MATA23 Linear Algebra I
Date: April 16th , 2016
Duration: 180 minutes
Examiner: Sophie Chrysostomou
Xiamei. Jiang
1. (10 points)
a) Use Cramers rule to solve the linear system
x + 2y + z = 4
3x + y 2z = 1
2x + 3y z = 9
1 2 1
3 1 =
A=
2
Linear Systems and Matrices
Linear Systems
DEFINITION: An m n linear system of equations is a system of m linear equations in n
variables:
a11 x1 + a12 x2 + . + a1n xn
= b1
a21 x1 + a22 x2 + . + a2n xn
= b2
.
.
.
.
. .
.
.
.
.
. .
am1 x1 + am2 x2 + . + am