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
MATH 310, REVIEW SHEET 1
These notes are a very short summary of the key topics in the book (and follow the book
pretty closely). You should be familiar with everything on here, but its not comprehensive,
so please be sure to look at the book and the lect
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
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
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
4.1 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
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
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