1.1
A Matrix 
row, columns 
Coefficients Matrix 
Just Left Hand Side 
Augmented Matrix 
Left and Right Hand Side 
Solving Linear Systems 
(1) Augmented Matrix (2) Row Operations (3) Solution to Linear System The RHS is the solution 
One Solution 
Upper triangle with Augmented Matrix 
No Solution 
Last row is all zeros = RHS number 
Infinitely Many Solutions 
Last row (including RHS) is all zeros 
Inconsistent 
Has No Solution 
1.2
Echelon Matrix 
(1) Zero Rows at the bottom (2) Leading Entries are down and to the right (3) Zeros are below each leading entry 
Reduced Echelon Matrix 
(1) The leading entry of each nonzero row is 1 (2) Zeros are below AND above each 1 
Pivot Position 
Location of Matrix that Corresponds to a leading 1 in REF 
Pivot Column 
Column in Matrix that contains a pivot 
To get to EF 
down and right 
To get to REF 
up and left 
Free Variables 
Variables that don't correspond to pivot columns 
Consistent System 
Pivot in every Column 
1.3
RR^{2} 
Set of all vectors with 2 rows 
1.4
Vector Equation 
x1a1+x2a2+x3a3 =b 
Matrix Equation 
Ax=b 
If A is an m x n matrix the following are all true or all false 
Ax = b has a solution for every b in RR^{m} Every b in RR^{m} is a lin. combo of columns in A Columns of A span RR^{m} Matrix A has a pivot in every row (i.e. no row of zeros) 
Anything in Bold means it is a vector.
1.5
Homogeneous 
Ax = 0 
Trivial Solution 
Ax = 0 if at lease one column is missing a pivot 
Determine if homogenous Linear System has a non trivial solution 
(1) Write as Augmented Matrix (2) Reduce to EF (3) Determine if there are any free variables(column w/o pivot) (4) If any free variables, than a nontrivial solution exists (5) NonTrivial Solution can be found by further reducing to REF and solving for x 
If Ax = 0 has one free variable 
Than x is a line that passes through the origin 
If Ax = 0 has two free variables 
Than x has a plane that passes through the origin 
1.7
Linear Independence 
No free Variables, none of the vectors are multiples of each other 
To check ind/dep 
reduce augmented matrix to EF and see if there are free variables(ie. every column must have a pivot to be linearly independent) 
To check if multiples 
u = c * v find value of c, then it is a multiple therefore linearly dependent 
Linearly Dependent 
If there are more columns than rows 
1.8
Every Matrix Transformation is a: 
Linear Transformation 
T(x) = 
A(x) 
If A is m x n Matrix, then the properties are 
(1) T(u + v) = T(u) + T(v) (2) T(cu) = cT(u) (3) T(0) = 0 (4) T(cu + dv) = cT(u) + dT(v) 
1.9
RR^{n} > RR^{m} is said to be 'onto' 
Equation T(x) =Ax=b has a unique solution or more than one solution each row has a pivot 
RR^{n} > RR^{m} is said to be onetoone 
Equation T(x) =Ax=b has a unique solution or no solution each row has a pivot 
2.1
Addition of Matrices 
Can Add matrices if they have same # of rows and columns (ie A(3x4) and B(3x4) so you can add them) 
Multiply by Scalar 
Multiply each entry by scalar 
Matrix Multiplication (A x B) 
Must each row of A by each column of B 
Powers of a Matrix 
Can compute powers by if the matrix has the same number of columns as rows 
Transpose of Matrix 
row 1 of A becomes column 1 of A row 2 of A becomes column 2 of A 
Properties of Transpose 
(1) if A is m x n, then A^{T} is n x m (2) (A^{T})^{T} = A (3) (A + B)^{T} = A ^{T} + B ^{T} (4) (tA)^{T} = tA^{T} (5) (A B)^{T} = B^{T} A^{T} 
2.2
Singular matrix 
A matrix that is NOT investable 
Determinate of A (2 x 2) Matrix 
det A = ad  bc 
If A is invertable & (nxn) 
There will never be no solution or infinitely many solutions to Ax = b 
Properties of Invertable Matricies 
(A^{1})^{1} = A (assuming A & B are investable) (AB)^{1} = B^{1} A^{1} (A^{T})^{1} = (A^{1})^{T} 
Finding Inverse Matrix 
[A  I ] > [ I  A^{1}] Use row operations STOP when you get a row of Zeros, it cannot be reduced 
2.3 Invertable Matrix Theorem


2.8
A subspace S of RR^{n} is a subspace is S satisfies: 
(1) S contains zero vector (2) If u & v are in S, then u + v is also in S (3) If r is a real # & u is in S, then ru is also in S 
Subspace RR^{3} 
Any Plane that Passes through the origin forms a subspace RR^{3} Any set that contains nonlinear terms will NOT form a subspace RR^{3} 
Null Space (Nul A) 
To determine in u is in the Nul(A), check if: Au = 0 If yes > then u is in the Nullspace 
2.9
Dimension of a nonzero Subspace 
# of vectors in any basis; it is the # of linearly independent vectors 
Dimension of a zero Subspace 
is Zero 
Dimension of a Column Space 
# of pivot columns 
Dimension of a Null Space 
# of free variables in the solution Ax=0 
Rank of a Matrix 
# of pivot columns 
The Rank Theorem 
Matrix A has n columns: rank A (# pivots) + dim Nul A (# free var.) = n 
dim = dimension; var. = variable
3.1
Calculating Determinant of Matrix A is another way to tell if a linear system of equations has a solution 
(1) Det(A) not =0, then Ax=b has a unique solution (2) Det(A) =0, then Ax=b has no solutions or inf many 
If Ax not= 0 
A^{1} exist 
If Ax = 0 
A^{1} Does NOT exist 
Cofactor Expansion 
Use row/column w/ most zeros 
If Matrix A has an upper or lower triangle of zeros 
The det(A) is the multiplication down the diagonals 
3.2
Determinate Property 1 
If a multiple of 1 row of A is added to another row to produce Matrix B, then det(B)=det(A) 
Determinate Property 2 
If 2 rows of A are interchanged to produce B, then det(B)=det(A) 
Determinate Property 3 
If one row of A is multiplied to produce B, then det(B)=k*det(A) 
Assuming both A & B are n x n Matrices 
(1) det(A^{T}) = det(A) (2) det(AB) = det(A)*det(B) (3) det(A^{1}) = 1/det(A) (4) det(cA) = c^{n} det(A) (5) det(A^{r}) = (detA)^{r} 
3.3 AKA Cramer's Rule
Cramer's Rule 
Can be used to find the solution to a linear system of equations Ax=b when A is an investable square matrix 
Def. of Cramer's Rule 
Let A be an n x n invertible matrix. For any b in RR^{n}, the unique solution x of Ax=b has entries given by xi = detAi(b)/det(A) i = 1,2,...n 
Ai(b) 
is the matrix A w/ column i replaced w/ vector b 
5.1
Au=λu 
A is an nxn matrix. A nonzero vector u is an eigenvector of A if there exists such a scalar λ 
To determine if λ is an eigenvalue 
reduce [(AλI)0] to echelon form and see if it has any free variables. yes > λ is Eigenvalue no > λ is not eigenvalue 
To determine if given vector is an eigenvector 
Ax=λx 
Eigenspace of A = 
Nullspace of (AλI) 
Eigenvalues of triangular Matrix 
entries along diagonal *you CANNOT row reduce a matrix to find its eigenvalues 
5.2
If λ is an eigenvalue of a Matrix A 
then (AλI)x=0 will have a nontrivial solution 
A nontrivial solution will exist 
if det(AλI)=0 (Characteristic Equation) 
A is nxn Matrix. A is invertible if and only if 
(1) The # 0 is NOT an λ of A (2) The det(A) is not zero 
Similar Matrices 
If nxn Matrices A and B are similar, then they have the same characteristic polynomial (same λ) with same multiplicities 
5.3
A matrix A written in diagonal form 
A=PDP^{1} 
Power of Matrix 
A^{k} = Diagonal matrix and #'s on diagonal get raised to the k 
Determining if Matrix is Diagonalizable 
λ of a nxn matrix 
n distinct (or real) λ then matrix is diagonalizable less than n λ, it may or may not be diagonalizable; it will be if # of linearly dependent eigenvectors = n 
eigenvectors of nxn matrix 
n linearly independent eigenvectors, then diagonalizable less than n linearly independent eigenvectors, then matrix is NOT diagonlizable 
D 
matrix w/ λ down diagonal 
P 
columns of P have linearly n linearly independent eigenvectors 
Finding P 
solve AλI and plug in the λ values. Reduce to EF, solve for x, & find eigenvector 


6.1
Length of vector x 
x = sqrt(x1^{2}+x2^{2}) 
Length fo vector x in RR^{2} 
x = sqrt(x • x) 
The Unit Vector 
u = v/v 
Two vectors u & v in RR^{n}, the distance between u & v 
u  v 
Two vectors u & v are orthogonal if and only if 
u+v^{2}= u^{2} +v^{2} u • v = 0 
6.2
The distance from y to the line through u & the origin 
z = y  yhat 
6.4
Gram Schmidt Process Overview 
take a given set of vectors & transform them into a set of orthogonal or orthonormal vectors 
Given x1 & x2, produce v1 & v2 where the v's are perp. to each other 
(1) Let v1=x1 (2) Find v2; v2=x2  x2hat 
x2 hat 
(x2•v1)/(v1•v1) * v1 
Orthogonal Basis 
{v1,v2,...,vn} 
Orthonormal Basis 
{v1/v1, v2/ v2,..., vn/vn} 
7.1
Symmetric Matrix 
A square matrix where A^{T}=A 
If A is a symmetric Matrix 
then eigenvectors associated w/ distinct eigenvalues are orthogonal If a matrix is symmetrical, it has an orthogonal & orthonormal basis of vectors 
Orthogonal matrix is a square matrix w/ orthonormal columns 
(1) Matrix is square (2) Columns are orthogonal (3) Columns are unit vectors 
If Matrix P has orthonormal columns 
P^{T}P=I 
If P is a nxn orthogonal matrix 
P^{T}=P^{1} 
A=PDP^{T} 
A must be symmetric, P must be normalized 

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luckystarr, 18:14 22 Apr 18
Won't download as PDF link is broken
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