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| Algebraic multiplicity for eigenvalue λ |
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| How many times it occurs in characteristic equation. |
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| It is the basis of the vectors that solve Ax = λx (so if x = (1,0,0) and (0,0,1) then the basis is {1,0,0},{0,0,1}) |
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| n x n matrix with all non negative entries and columns adding to one |
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| Let A be a Markov matrix. Then |
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| 1 is an eigenvalue of A and any other eigenvalue λ satisfies |λ|≤1 |
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| If A is a matrix with real entries and λ is a complex eigenvalue, |
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| The conjugate of λ is also a complex eigenvalue |
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| A linear system has either |
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| one unique solution or no solution or infinitely many solutions. |
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| Pivot variable vs free variable |
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Pivot variable is variable in pivot column
Free Variable is every other variable. |
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| A linear system is consistent if ___ |
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| Does not have a row like [0,0,0 | b] (b is a non zero number). |
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| Determine if w is in Span(u,v) (u,v,w are vectors)? |
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| Check if linear system corresponding to u,v|w is consistent (write as augmented form with vectors as columns) |
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| m x n matrix (columns, rows) |
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| Which rules apply for matrix multiplication? Which don't? |
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(a) A (BC) = (AB)C (associative law of multiplication)
(b) A (B + C) = AB + AC , (B + C) A = BA + CA (distributive laws)
AB != BA. Matrix multiplication is not commutative. |
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| elementary matrix, premutation matrix, and how they relate |
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Definition
elementary matrix - obtained by performing a single
elementary row operation on an identity matrix.
permutation matrix - obtained by performing row exchanges on an identity matrix |
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| Determine if a matrix A has LU decomp |
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| If A can be transformed into echelon form without the use of row exchanges, then A has LU factorization. |
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1) Find U using rowops
2) Take the opposite sign of row ops and insert correctly. |
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| Solve Ax = b using LU decomp |
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Definition
1) Solve Lc = b
2) Solve Ux = c |
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| Suppose A and B are invertible |
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Definition
AB is invertible.
(AB)^-1 = B^-1 A^-1
A^T and B^T are invertible |
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set of objects you can linear combine
(ex: matricies, polynomials) |
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H is a subset of V if:
It shares a 0 vector with V
Sums of things in H are in H
cU is in H if U is in H |
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| Nullspace (what it is and how to find) |
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Definition
Set of solutions to Ax = 0
To find
1) REF augmented matrix with 0 ([A|0)
2) Write solution as linear combination (ex:
(x1,x2,x3) = (1,2,3)x1 + ...
3) Nul(A) = Span (lin comb) |
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| Augment with a vector {b1,b2,b3} and see if consistent |
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| A single non-zero vector v1 is always linearly _____ |
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| Nul(A) and solutions to Ax = b |
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Definition
Let Axp = b
xp + Nul(A) will give all solutions in Nullspace. |
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| The columns of A are linearly independent means ___ (3 things) |
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Definition
Ax = 0 has only the solution x = 0.
Nul(A) = {0}
A has n pivots |
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| Vectors v1, . . . , vp containing the zero vector are linearly ____ |
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Definition
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| A set of vectors{v1, . . . ,vp} in V is a basis of V if |
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Definition
V = span(vectors)
vectors linearly independent |
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| it has a basis of p vectors |
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| To be a basis of R^n the set must |
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| The dimension of V is the |
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| number of elements in the basis |
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| A basis for Col(A) is given by |
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Definition
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| Find T with respect to std basis |
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Definition
[T(e1) T(e2) T(e3)] where T(e1), T(e2), ... are the columns of the matrix.
Example:
[image]
[image] |
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Definition
1) Obtain each vector in B as a linear combination of C.
2) Use the "coordinates" from your combination as columns.
[T(x1)_C, T(x2)_C, ...] |
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| same as dot product betweeen 2 vectors (ex: v transpose w) |
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| vectors that are unit vectors and orthogonal |
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| space of all vectors that are orthogonal to the subspace W |
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| Nul(A) is the orthogonal complement of |
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| Col(A) is the orthogonal complement of |
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| Find all vectors orthogonal to v1 and v2 |
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Definition
Find the orthogonal complement of Col(v1 v2)
Can orthogonal complement using Nul(A^T) |
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Definition
Dimension: m edges x n nodes
A(i,j)=
−1, if edge i leaves node j
+1 if edge i enters node j
0 otherwise |
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| Meaning of nullspace of incidence matrix |
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Definition
| dim(Nul(A)) is number of connected subgraphs |
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| Meaning of left nullspace of incidence matrix |
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Definition
left nullspace = null(A^T)
dim(Nul(A^T)) is # of independent loops |
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| Orthogonal basis definition |
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Definition
| If the vectors are orthogonal to each other. |
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| Orthogonal projection of vector x ONTO vector y |
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Definition
Used to project x into y by using Px
Find using: [image] |
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| Orthogonal projection of x onto W |
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Definition
Determined by xHat
[image]
Once xˆ is determined, x⊥ (error term) = x − xˆ. |
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| Closest point to x in span(v1,v2,..) |
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Definition
Follow this formula. Resulting vector/pt is closest)
[image] |
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| Find least squares solution to Ax = b (and define meaning of soln) |
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Definition
Solve [image]
[image] is minimal |
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| Projection matrix for proj onto Col(A) |
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Definition
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Definition
Solve for beta:
[image]
where:
[image]
[image]
[image]
Line is [image] |
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| The columns of Q are orthonormal means ___ |
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Definition
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| Given a basis a1, . . . , an, produce a orthogonal basis b1, . . . , bn and an orthonormal basis q1, . . . , qn. |
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| Least square solution using QR decomp (Ax = b) |
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Definition
Find QR decomp.
Rx = Q^T b, x will be best sol'n |
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| Determinant of matrix (non std way) |
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Definition
1) Get into upper triangular matrix
2) multiply diagonal |
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Definition
1) Gram-Schmidt on columns of A to get columns of Q
2) R = Q^T*A
Q is orthonormal, R is upper triangular |
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Definition
I_C,B x_b
x_b is the coordinate vectors of adding from basis b.
I_C,B is special case of T_C,B where I(v) = v |
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| 3x3 matrix with detA = 5 det(2A) = ? |
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Definition
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Definition
| Ax = λx (x is a eigenvector, λ is eigen value) |
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| Solve for eigenvalues (λ) of matrix A |
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Definition
| det(A - λ * Identity) = 0 |
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Diagonal matrix A^100
(a,0,0)
(0,b,0) = A
(0,0,c) |
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Definition
(a^100,0,0)
(0,b^100,0)
(0,0,c^100) |
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| Diagonalize A and find A^n |
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Definition
1) Find eigenvectors
2) Find P = I_e,b (e is std basis, b is eigenvector basis)
3) Find D matrix using eigenvalues (eigenvalues down diagonal)
4) A^n = P D^n P^-1 (where D is diagonal) |
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| An n × n matrix A is diagonalizable if and only if _____ |
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Definition
| A has n linearly independent eigenvectors |
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Definition
| If A is symmetric (A = A^T), then it has an orthonormal basis of eigenvectors and all eigenvalues are real |
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Sigma not eigenvalues (eigenvalues of A^T A not A)
SVD not unique. |
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