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,
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
FINAL EXAMINATION
MATA23H
Linear Algebra I
Examiner: N. Cheredeko
Date: April 14, 2004
Duration: 3 hours
I. Multiple Choice Questions
Display the answers in the front page
18.06 Professor Postnikov Quiz 3 November 23, 2009
SOLUU 0N3
Your PRINTED name is:
Please circle your recitation:
(R01) T10 2132 HwanChul Yoo Gradmg
(302) T11 2-132 HwanChul Yoo
(R03) T12 2132 David Shirokoff 1
(R04) Tl 2131 Fucheng Tan
(R05) T1 2-
18.06 Quiz 3 Solutions
sor Strang
May 8, 2010
Profes-
Your PRINTED name is:
1.
Your recitation number is
2.
3.
1. (40 points) Suppose u is a unit vector in Rn , so uT u = 1. This problem is about the n by n
symmetric matrix H = I 2u uT .
(a) Show directly
18.06 Quiz 2
April 7, 2010
Professor Strang
Your PRINTED name is:
1.
Your recitation number or instructor is
2.
3.
1. (33 points)
(a) Find the matrix P that projects every vector b in R3 onto the line in the direction of
a = (2, 1, 3).
Solution The genera
18.06 Quiz 1
March 1, 2010
Professor Strang
Your PRINTED name is:
1.
Your recitation number or instructor is
2.
3.
4.
1. Forward elimination changes Ax = b to a row reduced Rx = d: the complete solution is
4
2
5
x = 0 + c1 1 + c2 0
0
0
1
(a) (14 points)
University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
FINAL EXAMINATION
MATA23H3
Linear Algebra I
Examiner: S. Chrysostomou
Date: April 13, 2011
Duration: 3 hours
FAMILY NAME:
GIVEN NAMES:
STUDENT NUMBER:
SIGNATURE:
DO NOT OP
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 component
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
Denition: Let A = ai
Mn,k (R) and B = b
j
Mk,m(R).
The matrix product C = AB is the n m matrix C = ci j
where
k
ai b j = ai1 b1j + ai2 b2j + + aik bkj = ai bj .
ci j =
=1
(ai is the ith row of A and bj is the j th column of B .)
Denition: The n n identi
Denition: Any matrix that can be obtained from an identity matrix by means of one elementary row operation is called an elementary matrix.
Theorem: Let A Mn,k (R) and let E Mn,n(R) be elementary.
Multiplication of A on the left by E eects the same element
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,
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
Let a = [a1, a2, a3], b = [b1, b2, b3] R3. Then the vector
a2 a3
e1
e3
e2 +
b1 b3
b2 b3
a1 a2
a1 a3
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.
Pr
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
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
University of Toronto at Scarborough
Department of Computer and Mathematical Sciences
Midterm Test
MATA23 - Linear Algebra I
Examiner: R. Grinnell
Date: June 20, 2012
Time: 5:00 pm
Duration: 120 minutes
Provide the following information:
(Print) Surname:
SOLUTIONS TO QUIZ 3
Problem 1. (6 points each)
a
c + di
A=
c di
b
a) This matrix is clearly hermetian.
b) Thus, the two eigenvalues are real.
c) The sum of the eigenvalues is tr(A) = a + b.
d) The product of the eigenvalues is det (A) = ab (c + di)(c di)
18.06
QUIZ 2
April 06, 2007
Your PRINTED name is: SOLUTIONS
Please circle your recitation:
(1)
M2
2-131 A. Osorno
(2)
M3
2-131 A. Osorno
(3)
M3
2-132 A. Pissarra Pires
(4) T 11 2-132 K. Meszaros
(5) T 12 2-132 K. Meszaros
(6)
T1
2-132 Jerin Gu
(7)
T2
2-13