midterm 2 Flashcards
let u, v, and w be vectors in R^n, and let c be a scalar. Then, _________________ (properties of the inner product)
u * v = v * u
(u + v) * w = u * w + v * w
(cu) * v = c(u * v) = u * (cv)
u * u >= 0 +, and u * u = 0 in and only if u = 0
the length (or norm) of v is the nonnegative scalar ||v|| defined by ______________
||v|| = √(v * v) = √(v1^2 + v2^2 + … Vn^2)
and
||v^2|| = v * v
for u and v in R^n, the distance between u and v, written as dist(u, v) is the length of the vector u - v. That is, ______________
dist(u, v) = ||u - v||
two vectors u and v in R^n are orthogonal (to each other) if __________
u * v = 0
two vectors u and v are orthogonal if and only if __________________
||u + v||^2 = ||u^2|| + ||v^2||
how do you check to see if a set of vectors is an orthogonal set?
check the dot product between each of the vectors
if S is an orthogonal set of nonzero vectors in R^n, then S is _________________
linearly independent and a basis for the subspace spanned by S
an orthogonal basis for a subspace W of R^n is a _________________
basis for W that is also an orthogonal set
let {u1 ….. up} be an orthogonal basis for a subspace W or R^n. For each y in W, the weights in the linear combination y = c1u1 + … + cpup are given by ________________
cj = (y * uj) / (uj * uj)
a set is an orthonormal set if ________________
it is an orthogonal set of unit vectors
how do we check if a set is orthonormal?
1) check if the set is orthogonal
2) check if the lengths of each of the vectors is 1
an mxn matrix U has orthonormal columns if and only if ________
U^T * U = I
what is a vector space?
a nonempty set of V objects, called vectors, on which are defined by two operations, called addition and multiplication by scalars, subject to the ten axioms. The axioms must hold for all vectors u, v, and w in V and for all scalars c and d
what are the vector space axioms?
1) the sum of u and v, denoted u + v, is in V
2) u + v = v + u
3) (u + v) + w = u + (v + w)
4) for every vector u in V, there is a zero vector in V such that u + 0 = u
5) for every vector u in V, there is a vector -u in V such that u + (-u) = 0
6) the scalar multiple of u by c, denoted cu, is in V
7) c(u + v) = cu + cv
8) (c + d)u = cu + du
9) (cd)u = c(du)
10) 1u = u
a subspace of a vector space V is a subset H of V that has __________
three properties
what are the properties to be a subspace of a vector space?
1) the zero vector of V is in H
2) H is closed under vector addition. This means for each vector u and v in H, the sum u + v is in H
3) H is closed under scalar multiplication. This means for each vector u in H and every scalar c, the vector cu is in H
the null space of an mxn matrix A, written as Nul A, is the _______________________
set of all solutions to the homogeneous equation Ax = 0.
what is nullspace in set notation?
Nul A = {x : x is in R^n and Ax = 0}
the column space of an mxn matrix A, written as Col A, is _________________
the set of all linear combinations of the columns of A
If A = [a1 … an], the column space of A is ________
Col A = Span{a1, … an}
or in set notation
Col A = {b : b = Ax for some x in R^n}
the row space of an mxn matrix A, written as Row A, is __________________
the set of all linear combinations of the row vectors in A
if v1 … vp are in a vector space V, then _______
Span{v1 … vp} is a subspace of V
the set of all solutions to the system Ax = 0 of m homogeneous linear equations and n unknowns (null space) is _______________
a subspace of R^n
let H be a subspace of a vector space V. A set of vectors B in V is a basis for H if ______
i) B is a linearly independent set
ii) the subspace spanned by B coincides with H, meaning H = Span B
if A is an invertible matrix,
the columns of A form a basis for R^n because they are a linearly independent set and span R^n
the pivot columns of a matrix A _________
form a basis for Col A
if two matrices are row equivalent, __________
then their row spaces are the same
if two matrices are row equivalent and B is in echelon form, ___________
the nonzero rows of B form a basis for the row space of A and B
if a vector space V has a basis B = {b1, … b2}, then any set in V __________________
containing more than n vectors must be linearly dependent
if a vector space V has a basis of n vectors, ____________
then every basis of V must consist of exactly n vectors
if a vector space V is spanned by a finite set, __________________________
then V is said to be finite-dimensional and the dimension of V, written as dim V, is the number of vectors in a basis for V.
let V be a p-dimensional vector space.
any linearly independent set of exactly p elements in V is automatically a basis for V.
any set of exactly p elements that spans V is automatically a basis for V
the rank of an mxn matrix A is ___________
the dimension of the column space of A
the nullity of an mxn matrix A is _________
the dimension of the null space of A
the dimensions of the column space and the null space of an mxn matrix A satisfy the equation _____________
rank A + nullity A = number of columns in A
let A be an nxn matrix. the following statements are equivalent (invertible matrix theorem)
m) the columns of a form a basis of R^n
n) Col A = R^n
o) rank A = n
p) nullity A = 0
q) Nul A = {0}
what is the definition of a matrix factorization?
an equation that expresses a matrix A as a product of two or more matrices
in LU factorization, what is U?
the echelon form of A
in LU factorization, what is L?
an mxm lower triangular matrix with 1s on the diagonal, the other entries come from the row operations done to get a into echelon form (to make U)
what is Col A?
the span of the columns of matrix A that correspond to the pivot positions in the REF of A
what is Nul A?
the vectors corresponding to the parametric equation coming from Ax = 0
what is a coordinate vector?
a vector that has entries that are the weights (c values) of the linear combination of a specific vector
when selecting a basis for a subspace, choose vectors that
span H
are linearly independent
what is the definition of nullity?
the number of vectors in the basis of Nul A
what is the definition of rank?
number of nonzero rows in echelon form
the dimension of Row A is the same as
the rank of A
the dimension of Col A is the same as
the rank of A
the dimension of Nul A is
the number of free variables in the equation Ax = 0
an eigenvector of an nxn matrix A is a ____________
nonzero vector x such that Ax = λx
a scalar λ is called an eigenvalue of A if _____________
there is a nontrivial solution x for Ax = λx
what is the definition of eigenspace?
the nullspace of the matrix A - λI. the eigenspace contains the zero vector and all eigenvectors corresponding to λ
all the eigenvalues of a given matrix satisfy the equation ________
det(A - λI) = 0
an nxn matrix is diagonalizable if and only if ______________
it has n linearly independent eigenvectors
the power method is used to estimate _____________
the dominant eigenvalue of a matrix
what is a probability vector?
a vector with nonnegative entries that add up to 1 (all entries are between 0 and 1)
what is a stochastic matrix?
a matrix that holds probability vectors
if P is a stochastic matrix ____________
1 is an eigenvalue of P
a steady state vector for a stochastic matrix P is a vector q such that ______
Pq = q
to find a steady state vector _________
find the eigenvector associated with the eigenvalue 1 and scale it to be a probability vector
a stochastic matrix is regular if there exists a positive integer k such that ____________
P^k has strictly positive entries (only need to find one k)
if P is an nxn regular stochastic matrix, then P has a _______________
unique steady state. the initial state has no effect on the long-term outcome
the inner product (dot product) u * v is ____
u^T * v
a unit vector is a vector _________
of length 1
an nxn matrix has orthonormal columns if and only if _______
U^T * U = I
what is the definition of an orthogonal matrix?
a square matrix U such that U^T = U^-1 and U has orthonormal columns <– implied by saying U is orthonormal matrix
what is an orthogonal matrix?
a square matrix with orthonormal columns
what is the proof for an orthonormal matrix? (U^T * U = I)
1) U^T * U = I
2) (U^T * U) * U^-1 = I * U^-1
3) U^T(UU^-1) = U^-1
4) U^T * I = U^-1
5) U^T = U^-1
what is a symmetric matrix?
a square matrix such that A^T = A^-1
an nxn matrix is orthogonally diagonalizable if and only if _______
A is symmetric
if A is symmetric, then any two eigenvectors from different eigenspaces are _______
orthogonal
if A is symmetric, what does this tell us about matrix P is A = PDP^-1?
P^-1 = P^T, because P is an orthogonal matrix
when is the length of Ax maximalized?
when x = v1 (the largest eigenvalue’s corresponding eigenvector)
the singular values of a matrix A are the ___________________
square roots of the eigenvalues of A^T*A, arranged in decreasing order (sigma 1 is the largest)
what do the singular values of a matrix represent?
the lengths of the vectors Av1, Av2, …, Avn
what is Σ in the SVD equation?
an mxn (same size as original matrix) block matrix of the form Σ = [ D 0, 0 0]
what is D in the SVD equation?
an rxr (r = rank = 3 of nonzero columns of A in echelon form) matrix with the diagonal entries being the first r singular values of A
what is the matrix U in the SVD equation?
an mxm orthogonal matrix where the vectors of U are un = (1/σn) * Avn
what is the matrix V in the SVD equation?
an nxn orthogonal matrix where the vectors of V are the eigenvectors of A^T*A
what is the SVD equation?
A = UΣV^T
