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Linear Algebra Final
| Term | Definition |
|---|---|
| Meaning of an mxn linear system | There are m equations with n unknowns |
| An mxn system where m > n | Overdetermined system |
| An mxn system with m < n | Underdetermined system |
| An mxn system with m = n | Square system |
| Possible solution set for a linear system | No solution, a unique solution, or infinite solutions |
| When there exists a solution to a linear system | Consistant |
| When there doesn't exist a solution to a linear system | Inconsistent |
| Row echelon form | All zero rows are at the bottom, the first nonzero term in a nonzero row is 1, numbers below leading ones is 0 |
| Reduced row echelon form | A matrix is in REF and the numbers above leading ones is 0 |
| Possible solution sets for underdetermined system | No solution or infinite solutions, not possible for a unique solution |
| Lead variable | Variable with a leading one, set equal to a constant or free variables |
| Free variable | Variable without a leading one |
| Homogenous system | System with the right side being entirely 0s, is always consistent |
| Matrix equality | If two matrices share the same dimension and same entries |
| Matrix addition | Possible with two mxn matrices and results in a mxn matrix, add up each corresponding entry |
| Scaler multiplication | Multiply each entry by the corresponding scaler |
| Zero matrix | Matrix consisting entirely of zeroes, the additive identity |
| Linear combination | The adding of numbers times different constants, ex c1a1 + ... + cnan = b1 |
| Theorem: Ax=b is consistent if and only if b can... | ...be written as a linear combination of the column vectors of A |
| A special type of matrix that extends out in one direction, can be either a row or column vector | Vector |
| Matrix multiplication | Can be done with an mxn and nxr matrix, A and B, results in a mxr matrix. Each entry is defined as sum(k=1 to n) of (a(ik) * b(kj)). Note: is NOT commutative |
| Matrix transpose | For an mxn matrix, results in nxm, flips the horizontal and vertical positions of each entry. |
| If a matrix equals its own transpose | Symmetric matrix |
| A square matrix with 1s along the diagonal and zeroes elsewhere, multiplicative identity, I | Identity matrix |
| If a matrix, A, has an inverse, B, s.t. AB=BA=I | Invertible matrix |
| Steps to find a matrix inverse | Augment a matrix with the identity matrix, put a matrix in RREF, resulting matrix on the right is the inverse |
| A matrix that is the identity matrix with one row operation done on it | Elementary matrix, usually put as E(i) |
| Type I elementary matrix | Swaps two rows |
| Type II elementary matrix | Scale a row by a nonzero constant |
| Type III elementary matrix | Add a row scaled by a constant to another |
| If A = E(1)E(2)...E(k), then A(-1) =... | E(k)(-1) * E(k-1)(-1) * ... * E(1)(-1) |
| Matrix that is either upper or lower triangular | Triangular matrix |
| Matrix with zeroes below the diagonal | Upper triangular matrix |
| Matrix with zeroes above the diagonal | Lower triangular matrix |
| A matrix with zeroes in every entry except the diagonal, is both upper and lower triangular | Diagonal matrix |
| LU factorization | Perform type III row operations, keep track of scalers used. Resulting matrix is U and matrix with 1/(scaler) below diagonal is L |
| Determinant of a 2x2 matrix, A | a11*a22 - a12*a21 |
| Determinent of nxn matrix, A | sum(k=1 to n) of akn * det(B) where B is the resulting matrix of removing column and row akn is in |
| Meaning of det(A) | If it is not zero then A is nonsingular, otherwise A is singular |
| Thm: det(A*B) = ... | det(A) * det(B) |
| Determinant of type I elem. matrix | det(E) = -1 |
| Determinant of type II elem. matrix with scaler b | det(E) = b |
| Determinant of type III elem. matrix with scaler b | det(E) = 1 |
| A set of vectors closed under scaler multiplication and vector addition | Vector space |
| Vector space C[a,b] | Set of all real valued, continuous function on [a,b] |
| Vector space Pn | Set of all polynomials of degree n-1 |
| A nonempty subset of a vector space | Subspace |
| A subspace formed by all the linear combination of v1, ..., vn | Span of v1, ..., vn |
| Spanning set | A set of vectors part of a vector space where their span equals the vector space |
| Null space of matrix A | The subspace containing the set of solutions to the homogenous system of A, defined as N(A) |
| Linearly independent/dependent | If c1*v1, .., cn*vn = 0, then c1=...=cn=0 is the only solution |
| Thm: Matrix A being nonsingular is equivalent to the column vectors of A being... | The column vectors of A are linearly independent |
| Thm: u ∊ Span(v1, ..., vn) has a unique linear combination if and only if... | If and only if v1, ..., vn are linearly independent |
| Basis of vector space V | A set of vectors that are linearly independent and span V |
| Thm: if v1, ..., vn and u1, ..., um both span V where m > n, then... | u1, ..., um are linearly dependent |
| Thm: If v1, ..., vn and u1, ..., um are both bases for V, then... | Then n = m |
| Dimension of vector space V, dim(V) | The number of vectors required to form a basis for V, except for the zero vector space which is 0 |
| Thm: If dim(V) = n > 0 for vector space V, then... | 1. Any m number of linearly independent vectors where m < n can be extended to form a basis for V 2. Any spanning set of k > n vectors can be pared down to form a basis for V |
| Coordinate vector | If 𝓥 = {v1, ..., vn} is a basis for V and some vector v = c1v1 + ... + cnvn, then (c1, ..., cn) is a coordinate vector of v with respect to 𝓥, [v]𝓥 |
| Transition matrix from Ų to 𝓥 | Converts a vector with respect to Ų to the same vector with respect to 𝓥 |
| Linear transformation, L, mapping from V to W | Function that converts to some v ∈ V to some v ∈ W L(c*v) = c*L(v) L(v1 + v2) = L(v1) + L(v2) |
| Kernel of linear transformation L:V->W | Subspace of V of vectors s.t. L(v) = 0 |
| Image of linear transformation L | Subspace of W consisting of the range of V, all possible outputs |
| Row space of mxn matrix A | Subspace of ℝ1xn spanned by the row vectors of A, Row(A) |
| Column space of mxn matrix A | Subspace of ℝ spanned by the column vectors of A, Col(A) |
| Thm: two row equivalent matrices share the same... | Two row equivalent matrices share the same row space |
| Row equivalent matrices | Two matrices where one matrix can be made into the other from row operations |
| Rank of matrix A | Dimension of the row/column space of A, rank(A) |
| Thm: the dimension of the row space of A equals... | Equals the dimension of the column space of A |
| Nullity of matrix A | Dimension of the null space of A, null(A) |
| Rank-Nullity theorem | If A is an mxn matrix, then row(A) + null(A) = n |
| Matrix representation theorem | Let L:V -> W and let dim(V) = n and dim(W) = m. There exists some mxn matrix A s.t. L(x) = Ax for all x ∈ V |
| Similarity theorem | If A and B are similar, then there exists some nonsingular matrix S such that B = S(-1)AS |
| Scaler product of x and y | x ᐧ y = x(T)*y |
| Norm of x | sqrt(x ᐧ x), is 0 if and only if x is the 0 vector |
| Distance between vectors x and y | norm(x - y) |
| Angle between two vectors | cos(θ) = (x ᐧ y)/(norm(x) * norm(y)) |
| Cauchy-Schwarz inequality | |x ᐧ y| <= norm(x) * norm(y) |
| Unit vector of x | Vector of length 1 in the same direction as x, e = (1/norm(x)) * x |
| Scaler production of x onto y | proj(x onto y) = (x ᐧ y)/norm(y) |
| Vector projection of x onto y | proj(x onto y) = (x ᐧ y)/(y ᐧ y) * y |
| Orthogonal subspaces X and Y | If for all x ∈ X and y ∈ Y, x ᐧ y = 0, written as X⟂Y |
| Orthogonal complement of X | The set of all vectors such that are perpendicular to every vector on X |
| Fundamental subspaces theorem | For every matrix A, the null space of A is equal to the orthogonal complement of the row space and the null space of A(T) is equal to the orthogonal complement of the column space |
| Least squares solution | If Ax=b is inconsistent there exists y that minimizes the residual |
| Residual of Ax=b | r(x) = norm(b - Ax) |
| Thm: Let S is a subspace of ℝm. For each b ∈ ℝm, there exists... | There exists some p ∈ S that is closest to b |
| Normal equation for Ax = b | A(T)*Ax = A(T)*b |
| Inner product for inner product space V | Operation between two vectors that satisfies the following for all u, v, w ∈ V: 1. <v, v> >= 0 and equality if and only if v = 0 2. <u, v> = <v, u> 3. <cu, v> = <u, cv> = c<u, v> 4. <u + v, w> = <u, w> + <v, w> |
| Pythagorean Law | If u⟂v, then norm(u + v)^2 = norm(u)^2 + norm(v)^2 |
| Orthogonal set for inner product space | If every vector on the set are orthogonal to another vector |
| Orthonormal set for inner product space | If the set of vectors are orthogonal and ever vector is a unit vector |
| Thm: If {u1, ..., un} is an orthonormal basis for IPS V and v = c1u1 + ... + cnun, then... | Then ci = <v, ui> for 1 <= i <= n |
| Coro: Let {u1, ..., un} is an orthonormal basis for IPS V. If u = a1u1 + ... + anun and v = b1u1 + ... + bnvn, then... | Then <u, v> = a1b1 + ... + anbn |
| Parseval's Theorem | If {u1, ..., un} is an orthonormal basis for IPS V and v = c1u1 + ... + cnvn, then norm(v)^2 = c1^2 + ... + cn^2 |
| Orthogonal matrix | If the column vectors of matrix Q are orthonormal, then Q is orthogonal |
| Properties of orthogonal matrix | The matrix's inverse is its transpose, Q* Q(T) = I <Qx, Qy> = <x, y> norm(Qx)^2 = norm(x)^2 Multiplying the matrix by a vector preserves the vector's length |
| Thm: If matrix A is orthogonal, then the least squares solution for it is... | x = A(T)b |
| Gram-Schmidt process for {x1, ... ,xn} | Define u1 = (1/norm(x1)) * x1 Then for 1 <= k <= n - 1 Define pk = <x(k+1), u1>u1 + <x(k+1), u2>u2 + ... + <x(k+1), uk>uk Define uk = (x(k+1) - pk)/norm(x(k+1) - pk) The resulting orthonormal basis is {u1, ..., un} |
| Eigenvalue and eigenvector of A | If Ax=λx, then λ is an eigenvalue and x is an eigenvector |
| Way to solve for eigenvalues/vectors for A | Find the values of λ that solve the roots for det(A - λI) = p(λ), p(λ) is the characteristic polynomial To solve for x, find the null space of A-λI after solving for λ |
| Equivalency theorem for eigenvalues for matrix A | 1. λ is an eigenvalue with eigenvector x 2. (A-λI)x = 0 has a nontrivial solution 3. N(A - λI) \=\ {0} 4. A - λI is singular 5. det(A - λ |
| Eigenvalues of a triangular matrix | The diagonal entries for the matrix |
| Thm: If an nxn matrix A has eigenvalues λ1, ..., λn, then det(A) = ... | det(A) = λ1*...*λn |
| Diagonalizable matrix | A is diagonalizable if there exists a nonsingular matrix X and a diagonal matrix D s.t. D = X(-1)*A*X X is said to diagonalize A If A is not diagonalizable, it is defective |
| Thm: If λ1, ..., λn are distinct eigenvalues for an nxn matrix A with corresponding eigenvectors x1, ..., xn, then.... | Then the eigenvectors x1, ..., xn are linearly independent and A is diagonalizable |
| Thm: If A is diagonalizable, then the columns of X are... and the diagonal entries of D are... | The columns of X are the eigenvectors of A, and the corresponding diagonal entry D corresponding to a column of X are the eigenvalues of A |
| Thm: If A is diagonalizable, then A^k = ... | A^K = X * D^k * X(-1) |
| Norm of a complex a + bi | sqrt(a^2 + b^2) |
| Conjugate of a complex number a + bi | a - bi |
| Norm of a complex vector z = z1 + ... + zn | sqrt(norm(z1) + ... + (zn)) |
| Conjugate of a complex matrix | Take the conjugate of each entry |
| Hermitian of a matrix | Take the transpose of the conjugate of a matrix |
| Inner product on a complex vector space V, z, w, u ∈ V | 1. <z, z> >= 0 with equality if and only if z = 0 2. <z, w> = conj(<w, z>) 3. <cz + dw, u> = c<z, u> + d<w, u> |
| Standard inner product on ℂn for z, w ∈ ℂn | <z, w> w(H)z |
| Hermitian matrix | If for matrix A, A(H) = A |
| Unitary matrix | An nxn matrix that has column vectors form an orthonormal set in Cn |
| Spectral Theorem | If matrix A is hermitian, then there exists a unitary matrix U that diagonalizes A |
| Probability vector | Vector with entries that add up to 1 |
| Thm: if λ=1 is the dominant e-value for stochastic matrix A, then... | Then the markov chain with converge to a steady state vector |
| How to find the steady state vector with a transition matrix A | Find the basis vector of the null space of A - I |
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