i.
wmnuW ~ .-
Cramers Rule
Theorem, 1f Ax : b is a system of n linear' equations in n unknowns where .r :
detA 7 0, then the system has a unique solution given by
_ dot. A:
:t'
I detA '
where A
3 ve ofo F5
Example 4. /7 R1 . N 5 Exercises. Do what is asked.
3 L I . . .
1. Is S : cfw_ [ 21] , [g] , [765] linearly independent subset of R2? Nb . / 1. Find a basrs for R3 that Includes the vecto
Remark 10. Every matrix A E MWHCR) with full column rank (p(A) : n) has the following properties:
1. There are no free variables in the linear system At : b;
2. All columns of A are pivot columns;
3.
cfw_it [)1 Woh'hwmi '
. dot in '- WWW
grow: 8; \/ 91am M 5w
. Lesson 7 Linear Independence and Spanning Sets 2. If it happens that V : span S, then S is called the spanning set for V, or V is spanned
l
Lesson 10 OrthonormalBases and the Gram-Schmidt Process Denitions. Let V be a vector space with inner product ( , ) and 14,11 6 V.
- 1, We say u is orthogonal to u (n _L 1) whenever (u, v) : 0,
Deni
Math 114: Linear Algebra
Symmetric Matrices
Definition: An n n real matrix A satisfying A = AT is called a symmetric matrix.
Properties of Symmetric matrices
All roots of the characteristic polynom
Math 114: Homework 7
DIRECTIONS: You are allowed to use the Matlab to answer the questions in this homework. However,
you must present your solution in a complete and logical manner to get full credit
Math 114: Linear Algebra
Linear Independence of Vectors and Bases of a Vector Space
Recall: Let V be a vector space and S = cfw_v1 , . . . , vk V .
A vector v V is said to be a linear combination of
Math 114: Linear Algebra
Rank and Nullity of a Matrix
Rank and Nullity
Given an m n matrix A, define
The null space of A is N ul(A) = cfw_x Rn | Ax = 0. This is a subspace of Rn .
The column space o
Math 114: Homework 6
1
2
1
2 4 2
1. Determine if the set , , is linearly independent. Justify your answer.
2 5 1
3
9
1
2. Write DEP if the set of vectors is linearly dependent, otherwise, writ
o
i
l'
2. P2, the set of all polynomials of degree n S 2, is a subspace of 13
3. The set of all symmetric 2 X 2 matrices is a subspace of M2x2(R).
' Remark. We do not need to verify all properties
Remark 6. Let M , N be matrices which are compatible for multipication. Then
1. row rank of MN 5 row rank of N and Ktul) : M)
1' . 2. column rank of [MN 3 column rank of M.
Theorem. Let A E MmMOR), 1,
a:
Lesson 3 Inverse of a Matrix
Denitions.
1. An n X n matrix A is said to be nonsingular or invertible if there exists a unique n X n matrix B such that
AB = BA ; I". The matrix B is called the inver
. m wt
1 I 11 [zlr7KL'C/(KS
Example 8. Consider the homogeneous system 5 Find the value(s) of c for which the system
l ,
m+zg~3z3+2x4:0 1 1 73 2 ' avey = 1 C/ l
2x1 2m + 2:53 2 0 a 2 72 2 0 (coef
Therefore,
1 2 o 3 1
o 0 2 e
A Method for Finding the Inverse of a Nonsingular Matrix
1. Form the n X 271 augmented matrix [A : In].
4:1
1 3 :> ~1
~5 10 5
O 1 0 0
0 A? 1 0
1. 0 O 1
0 1
O 3
2 0
2
*2
~3
Lesson 5 Cofactor Expansion and Applications of the Determinant 1 1 0
Example 3. Given A = 2 3 4 . Let's [ind rst all of its nine col'actors.
Denition. If A is a square matrix, then the (i,j)-minor of
I A < 3 V T
. ' . . 49
a , ,
"t
I];
. 2. Interchange the top row with another row, if necessary, to bring a nonzero entry to the top of the column (111 (112 a1" aCI b1
found in Step 1. . I 6121 (122
.
Lesson 1 Matrices Properties and Operations
Definitions. .
1. An matrix with m rows and TL columns (an m by 77. matrix) A is a rectangular array of numbers:
N N
(111 (112 1
~21 (1-12 - ' ' -21:
fl =
74931
We 1
we 5
1- ,7.4.
See Kolmanls book for the proofs of the following properties:
Properties of the Determinant
1
2.
memes
\1
8.
9.
. detA : detAT
detA
det B = det A.
ows (or two columns) of A
i
l'
r
Lesson 2 Solutions of Linear Equations
Denitions.
. 1. An equation (11:51 + 11212 + + anzn : b, which expresses b in terms of the variables 11:5527 . . ,xn called
unknowns and constan
\E [4 atoms a V
some SFV
rm 17v.r~: - .
Lesson 8 - Basis and Dimensions
Denition. A set of vectors S : cfw_111,712, . . . ,vk in a vector space V is called a basis for V if
spans V that is, span S: V
.
Lesson 6 Vector Spaces and Subspaces
Denition. A vector :2 in R", or an nvector is the directed line segment denoted by
Ti
502
z:(cci,z2,.,rn)orz: , \vhcrexi,x2,.i,:cnlR.
Zn
Denition. The norm (
\(oum
a1+a3 : 072a1+a2 ~6a3 =0, 3(12 ~ 12% = O.
The system is equivalent to
3; a w vaxol tom
1 0 1 a1 0
W ninth ~
We martian WWW u NJMOC m o mu)
Example 9. Let V be the set of ordered pairs (:5, y) of
Math 114: Linear Algebra
Orthogonal Projections
Recall: Given a subspace W of a vector space V ,
1. dim(W ) + dim(W ) = n
2. If cfw_v1 , . . . , vk is a basis for W and cfw_vk+1 , . . . , vn is a b
Math 114: Linear Algebra
Matrix Representations of Linear Transformations
Coordinate Vectors
Recall: If B = cfw_v1 , . . . , vn is a basis for V and x V , then x can be written as a unique linear co
Math 114 Exercises Set 2
1. Let V be the set of all positive real numbers. Define the following operations:
i. u v = uv
ii. v = v
Prove that V is a vector space.
2. Explain why the following are not
Math 114 Exercises Set 3
1. Which of the following functions are linear transformations?
a. L : R3 R3 defined by L(x, y, z) = (x, y 2 + z 2 , z 2 )
b. L : R3 R3 defined by L(x, y, z) = (0, z, y)
c. L