Linear Algebra
Grinshpan
Problem set 6.1
1.
0, not invertible
5.
110, invertible
6.
9. 36, invertible
14.
no such k
19.
k = 0, 1
2, invertible
2.
24, invertible
10.
4.
0, not invertible
0, not invertible
8.
0, not invertible
7.
1, invertible
11. k =
1
2
9
Linear Algebra
Grinshpan
Problem set 2.3
1. Compute the matrix
1 0
0 1 a b ,
c d
0 0
1
2 1 2
3
products
1 0 1 1 2 3
0 1
1 3 2 1 ,
1 1 2 2 1 3
1
2
3 ,
1 0 1 2 1 ,
1 1
2. Find all matrices that commute with
3. For each matrix
this pattern to find
1
0
3.2 Properties of Determinants
Week 6, Wed 2/12/14
The following theorem tells how the determinant changes when row operations are performed:
Theorem 3 Let A be a square matrix.
(a) If a multiple of one row of A is added to another row of A to product a m
4.2 Null Space, Column Space and Linear Transformations
Rn is special vector space.
In this section, we talked about some of the subspace of Rn
Week 7, Wed 2/19/14
Null Space
Definition
The null space of an m n matrix A, written as Nul A, is the set of al
2.4 Partitioned Matrices
Week 6, Mon 2/10/14
Example 1. Write the Matrix A as the 23 partitioned (or blocked) matrix
Partitioned matrices can be multiplied use the usual row-column rule as if the blocks are
numbers, provided the partitioned of A and B are
3.1 Determinant
Week 6, Wed 2/12/2014
Notation: Aij is the matrix obtained from matrix A by deleting the ith row and jth column.
Definition. The (i, j) cofactor of A is the number Cij = (1) i + j det (Aij)
Example 1
A23 =
Common notation:
C23 =
is commonl
4.1 Vector Spaces and Subspaces
Week 6, Wed 2/12/2014
We can think of a vector space in general, as a collection of objects that behave as vectors do in Rn.
We also call these objects vectors.
Definition.
A vector space is a nonempty set V of objects, cal
2.5 LU Factorizations
Week 6, Mon 2/10/14
Definition: A factorization of a matrix A is to expresses A as a product of 2 matrices
Let A be m n. Assume A can be row reduced to REF (without ROW SWAPPING).
Then A = LU,
Where L is an m m square lower triangul
1.9 The Matrix of a Linear Transformation
Week 4, Mon 1/27/2014
We already know that every matrix transformation is a linear transformation
Conversely, every linear transformation is a matrix transformation x Ax. Why?
Identity matrix In
The i th column
2.1 Matrix Operations
Week 4, Wed 1/29/14
Matrix Notation:
Two ways to denote m n matrix A:
In terms of the columns of A:
A= [a1
a2
. an]
In terms of the entries of A:
= [a i j]
Main diagonal entries: _
Zero matrix 0, n n identity matrix In
Theorem 1:
Let
2.3 Characterizations of Invertible Matrices
Week 5, Wed Feb 5, 2014
Theorem 8 (The Invertible Matrix Theorem) (IMT)
Let A be a square n n. The following statement are equivalent (i.e for a given A,
they are either all true or all false)
a. A is an invert
2.2 Inverse of a Matrix
Week 4, Wed 1/29/14
The inverse of a real number 7 is denoted by or 7 1. We have 771 = 71 7 = 1
An n n matrix A is invertible (or nonsingular) if there is an n n matrix C satisfying
CA = AC = In
We call C the inverse of A. The inve
4.1 Vector Spaces and Subspaces
Week 7, Mon 2/17/14
We can think of a vector space in general, as a collection of objects that behave as vectors do in Rn.
We also call these objects vectors.
Definition.
A vector space is a nonempty set V of objects, calle
4.2 Null Space, Column Space and Linear Transformations
Rn is special vector space.
In this section, we talked about some of the subspace of Rn Week 7, Mon 2/17/14
Null Space
Definition
The null space of an m n matrix A, written as Nul A, is the set of al
4.3 Linear Independent Set; Bases
Week 7, Wed 2/19/14
Linear Independent Set
Definition
A set of vectors cfw_v1, ., v p in a vector space V is said to be linearly independent if the
vector equation
c1 v1 + c1 v1 +.+ c1 v p = 0
has ONLY the trivial solutio
Linear Algebra
Grinshpan
3/2
1/2
1.
3/2
1/2
Problem set 2.2
3. arcsin
followed by stretching by 2.
4
1/3 1/3 1/3
0
1
4. 1/3 1/3 1/3
5.
1 0
1/3 1/3 1/3
2. Clockwise rotation by
3
4
= arccos 2.498
5
5
6. A(Ax) = x, Av = v, Aw = w, v w, v is on the line of
Linear Algebra
Grinshpan
Problem set 2.3
a b
1 1
0
1 2 3
ad bc
0
3
4 ,
0 1 , h
1. c d , 5
, 14, 2 4 6 ,
0
ad bc
0 0
6 2 4
3 6 9
a 0
a b
2.
.
.
0 d
b a
1 0
1 1001
0 1
3.
reflection,
horizontal shear,
reflection,
0 1
0
1
1 0
1 1 1
1
1 1
orthogonal project
Linear Algebra
Grinshpan
Problem set 2.2
1. Find the matrix of rotation by an angle of 60 in the counterclockwise direction.
1 1
2. Interpret the transformation T (x) =
x geometrically.
1 1
.8 .6
3. The matrix
represents a rotation. Find the angle of rota
Linear Algebra
Grinshpan
Problem set 1.1
1. Solve
4x + 3y = 2
using elimination. Check your answer.
7x + 5y = 3
x + 2y + 3z = 8
x + 3y + 3z = 10 using elimination. Check your answer.
2. Solve
x + 2y + 4z = 9
3. The sums of any two of three numbers are 2
Linear Algebra
Grinshpan
Problem set 2.1
1. Which of the following transformations are linear? Explain your answer.
y1 = x2 x3
y1 = 2x2
y1 = 2x2
y2 = x1 x3
y2 = 3x3 ,
y2 = x 2 + 2 ,
y = x x .
y = x
y = 2x
3
1
2
3
1
3
2
1
0
0
7
6
13
2. Find th
Li Sheng, Linear Algebra (math201), Lay 4th Ed Winter14
Week 1, Mon Jan 6, 2014
1.2 Row Reduction and Echelon Form
Echelon Form
1. Nonzero rows are above rows of all 0s
2. Each leading entry (i.e leftmost nonzero entry) of a row is to the right of the lea
Review for Exam 1 (1.1 2.3)
Let T: RnRm be a linear transformation, and let A be its standard matrix. Fill in the spaces, and explain.
(A) A is an _ by _ matrix
(B) Columns of A are linearly independent if and only if Ax = 0 has _ the trivial solution.
(C
Review for Exam 1 (1.1 2.3)
Week 5 Mon 2/3/2014
Let T: RnRm be a linear transformation, and let A be its standard matrix. Fill in the spaces, and explain.
(A) A is an _ by _ matrix
(B) Columns of A are linearly independent if and only if Ax = 0 has _ the
Linear Algebra (math201), Li Sheng
Week 3, changed to week 4
Quiz 2 (1.5, 1.7, 1.8, 1.9)
Name (Print):
Show your work. You will not get the full credit if you do not show your work.
1. (3pts) Describe all solutions of Ax b in (parametric) vector form, w
1 2 1 4 0
1 2 1 4 0
(1) (4.3#14w13) Given A = 2 4 4 6 4 row equivalent to B = 0 0 2 2 4.
3 6 5 10 4
0 0 0 0 0
(A) Find a such that Nul A is a subspace of Ra. Solution: a = _
(B) Find b such that Col A is a subspace of Rb. Solution: b = _
(C) Find a basis
1.8 Introduction to Linear Transformation
Week 2, Wed 1/15/2014
Recall functions
f (x) = x2
f (x1, x2) = x1 + x2
f (x1, x2, x3) = x12 + x22 + x32
f :domain co-domain:
Range:
Example 1.
A
x
If A is an m n matrix, we can think of Ax as a function (or mappin
1.7 Linear Independence - continue
Week 2, Wed 1/15/2014
Definition
A set of vectors cfw_v1, v2, , vp in Rn are linearly independent if the vector equation
x1v1 + x2 v2 + , , + xp vp = 0
has only the trivial solution (x1= x2 = = xp= 0).
A set of vectors c