MAT188 - THE BIG theorem
Let A = cfw_v1 , . . . , vn be a set of n vectors in Rm . Let A = [v1 . . . vn ] be an m n matrix. Let T : Rn Rm such that T (x) = Ax.
The followings are equivalent
The follo
MAT188 - Nov 6
1
Review: Properties of Determinant
The determinant associates each square matrix A a number, denoted by det A or |A|.
Denote Mij the submatrix of A by deleting the i-th row and j-col
MAT188 - Nov 2
1
Review: Determinant
The determinant associates each square matrix A a number, denoted by det A or |A|.
Denote Mij the submatrix of A by deleting the i-th row and j-column.
The mino
MAT188 - Oct 28
1
Review
A collection of k vectors cfw_v1 , . . . vk is a basis of S if it is linearly independent and it spans S.
All bases have the same number of vectors. The dimension of S is t
MAT188 - Oct 26
1
Review
Let S Rn . We say S is a subspace if (and only if)
0 S.
If u, v S, then u + v S.
If u S and s R, then su S.
A collection of k vectors cfw_v1 , . . . vk is a basis of S i
MAT188 - Oct 23
1
Review
Let S Rn . We say S is a subspace if (and only if)
0 S.
If u, v S, then u + v S.
If u S and s R, then su S.
Last time we talked about spancfw_v1 , . . . vk is a subspace
MAT188 - Oct 16
1
Review: Elementary Matrices
Recall that an elementary row operation corresponds to multiplying to the left by an elementary matrix.
The elementary matrix is given to be performing
MAT188 - Oct 19
1
Review
Let A be an n n matrix. Recall that A is invertible if there is a matrix B such that AB = BA = I.
B is called the inverse of A and it is denoted by A1 .
Also recall that a
MAT188 - Oct 21
1
Review
Last time we talked about subspaces in Rn . It is given by the form of the span of several vectors in
Rn .
They are innite, straight things, except for the zero subspace = s
MAT188 - Oct 14
1
Review: Linear transformations
A linear transformation is a multivarible vector-valued function T : Rn Rm satisfying the 2 linear
conditions:
For all x, y Rn , s R, we have T (x + y
MAT188 - Sept 30
1
Linear Independence
In Chapter 2.2, we talked about Span. Recall that the followings are equivalent:
cfw_v1 , . . . , vn span Rm
For any b Rm , b spancfw_v1 , . . . , vn
Every
MAT188 - Oct 9
1
Linear transformations
1 variable: A linear transformation f : R R is a function such that
.
for all x, y,
Remark: This is DIFFERENT from what you learnt before. Recall a linear fun
MAT188 - Sept 28
1
Exercises
1. True or False:
(a) If the echelon form of an augmented matrix of a system of linear equations has a [0 . . . 0|0] row, the
system has innitely many solutions.
TRUE / FA
MAT188 - Oct 5
1
Matrices
An m n matrix consists on m rows and n columns.
Addition and Scalar multiplication of matrices - Just like those with vectors
a11 + b11 . . . a1n + b1n
b11 . . . b1n
a
MAT188 - Oct 2
1
Theorems
If cfw_v1 , . . . , vk are linearly dependent, then
.
Proof : If cfw_v1 , . . . , vk are linearly dependent, that means there are xi s,
x1 v1 + . . . + xk vk =
Say xj is
.
MAT188 - Oct 30
1
Determinant function
A square matrix is invertible if there is another square matrix B such that AB = BA = I. Here
1 0 . 0
0 1 . . . 0
I = . . .
. .
.
. .
. .
.
0 0 . 1
is the ident
MAT188 - Nov 4
1
Review: Properties of Determinant
The determinant associates each square matrix A a number, denoted by det A or |A|.
Denote Mij the submatrix of A by deleting the i-th row and j-col
MAT188 - Ch2.2
1
System of linear equations
A system of linear equations is given by
a11 x1
.
.
.
am1 x1
+ .
.
.
+ .
+
a1n xn
.
.
.
=
b1
.
.
.
+ amn xn
=
bm
Here aij s and bj s are numbers and xj s
Review
1
Review
6.1. Let A be a square matrix. Then a nonzero vector v = 0 is an eigenvector of A if there exists a
scalar such that Au = u. Here is called an eigenvalue of A.
6.1. The eigenspace of
Review (Continued)
1
Short Questions
If A and B are n n matrices such that AB = 0, then show that rank(B) nullity(A). In particular, if
A2 = 0, then rank(A) n/2 and dim E0 n/2. (E0 is the eigenspace
MAT188 - Dec 2
1
Review
Two vectors u and v are orthogonal if their dot product is zero, i.e.
u v = u1 v1 + u2 v2 + . . . + un vn = 0.
Let S be a subspace in Rn . The orthogonal complement of S, den
MAT188 - Dec 4
1
Review
Let S be a subspace in Rn . The orthogonal complement of S, denoted by S , is the set of all the
vectors that is orthogonal to S, i.e.
S = cfw_v Rn such that v s = 0 for all s
MAT188 - Nov 30
1
Review
Two vectors u and v are orthogonal if their dot product is zero, i.e.
u v = u1 v1 + u2 v2 + . . . + un vn = 0.
Let S be a subspace with orthogonal basis cfw_v1 , . . . , vk .
MAT188 - Nov 27
1
Course evaluations
Please complete the course evaluations! Questions, comments, complaints, anything welcome!
TA evaluations.
2
Preliminaries - Orthogonality
Recall that in R2 , two
MAT188 - Nov 23
1
Review
Let T : Rn Rn be a linear transformation such that T (x) = Ax. v = 0 is an eigenvector corresponding
to an eigenvalue if Av = v.
If A has n eigenvalues (counting multiplicit
MAT188 - Nov 25
1
Review
Given an n n matrix A. If you have a system of linear dierential equations y = Ay, and A is
diagonalizable, then the general solution will be
y = a1 e1 x v1 + . . . + an en x
MAT188 - Nov 20
1
Review
Let B = cfw_v1 , . . . , vn be a basis for Rn . Then for any v Rn , v can be written as a linear combination
of the basis vectors,
a1
.
v = a1 v1 + . . . + an vn = [v1 . .
MAT188 - Nov 16
1
Review
Given a linear transformation T : Rn Rn , is an eigenvalue if there is a nonzero vector v such that
T (v) = Av. Such v is called an eigenvector corresponding to the eigenvalu
MAT188 - Nov 18
1
Review
Given a linear transformation T : Rn Rn , is an eigenvalue if there is a nonzero vector v such that
T (v) = Av. Such v is called an eigenvector corresponding to the eigenvalu
Faculty of Applied Science & Engineering, University of Toronto
MAT188H1F - Linear Algebra
Fall 2016
Tutorial Problems 4
1. Given a system of the linear equations, the last column of the augmented mat
Faculty of Applied Science & Engineering, University of Toronto
MAT188H1F - Linear Algebra
Fall 2016
Tutorial Problems 8
1. Suppose A is a 4 6 matrix that has reduced
1
0
R=
0
0
row-echelon form
0 1 1
UNIVERSITY OF TORONTO
FACULTY OF APPLIED SCIENCE AND ENGINEERING
First Year Program Core 8 and TrackOne
FIRST YEAR CORE 8 ENGINEERING PROGRAM
INTRODUCTION TO NUMERIC COMPUTATION
LABORATORY
MATLAB Over
Faculty of Applied Science & Engineering, University of Toronto
MAT188H1F - Linear Algebra
Fall 2016
Tutorial Problems 7
Recall that one reason for the word linear in linear mappings (transformations)
Faculty of Applied Science & Engineering, University of Toronto
MAT188H1F - Linear Algebra
Fall 2016
Tutorial Problems 6
2 1 1
2
x1
x1
1 (a) Let A = 2 3 2, y = 3, x = x2 , and B = 0
4 2 2
1
x3
0