1.1A 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 
1.2Echelon 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 
1.3RR^{2}  Set of all vectors with 2 rows 
1.4Vector 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.
1.5Homogeneous  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 
1.7Linear 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 
1.8Every 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) 
1.9RR^{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 
2.1Addition 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} 
2.2Singular 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
  2.8A 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 
2.9Dimension 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
3.1Calculating 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 
3.2Determinate 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 RuleCramer'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 
5.1Au=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 
5.2If 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 
5.3A 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 
  6.1Length 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 
6.2The distance from y to the line through u & the origin  z = y  yhat 
6.4Gram 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} 
7.1Symmetric 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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