## Linear Algebra Test 3

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etasler  on November 3, 2011

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# Linear Algebra Test 3

 a vector space, subset, the zero vector, u + v is in H, cu is in HA subspace of ________ V is a _________ H of V that has 3 properties. a. __________ of V is in H b. For each u and v in H, __________ c. For each u in H and each scalar c, __________
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#### English

a vector space, subset, the zero vector, u + v is in H, cu is in H A subspace of ________ V is a _________ H of V that has 3 properties. a. __________ of V is in H
b. For each u and v in H, __________ c. For each u in H and each scalar c, __________
the set of all solutions to the homogeneous equation Ax = 0 Definition: The null space of an m x n matrix, Nul A, is
Span {v1...vp} is a subspace of V If v1...vp are in a vector space V, then
the set of all linear combinations of the columns of A Definition The column space of an m x n matrix, Col A is
subspace of Rn The null space of an m x n matrix, Nul A is a
subspace of Rm The column space of an m x n matrix A is a
some vj (with j>1) is a linear combination of the preceding vectors v1...vj-1 An indexed set {v1... vp} of two or more vectors with v1 not zero is linearly dependent if and only if
linearly dependent A set containing the zero vector is
T(u + v) = T(u) + T(v) and T(cu) = cT(u) Linear Transformation from a vector space V into a vector space W is a rule that assigns to each vector x in V a unique vector T(x) in W such that for all u and v in x and all constants c
set of all vectors u in V such that T(u) = 0 The kernel of a linear transformation is the
set of all vectors in W of the form T(u) where u is in V The range of a linear transformation T from V to W is the
set of all linear combinations of the row vectors of a matrix A The row space, Row A, is the
has only the trivial solution A set of vectors v1...vp is said to be linearly independent if c1v1 + ... + cpvp = 0 ...
has a nontrivial solution A set of vectors {v1...vp} is said to be linearly dependent if c1v1 + ... + cpvp = 0...
c1v1 + ... + cpvp = 0 where c1...cp are not all zero A linear dependence relation is of the form
B is a linearly independent set and H = span {b1...bp} Let H be a subspace of a vector space V. The indexed set of vectors B= {b1...bp} in V is a basis for H if
column To test if the columns of a matrix are linearly independent look for a pivot in every
row To test if the columns of a matrix span Rm look for a pivot in every
the set formed from S by removing vk still spans H Let S= {v1 ... vp} be a set in V and let H = span {v1...vp}. If one of the vectors in S, say vk, is a linear combination of the remaining vectors in S, then
some subset of S is a basis for H Let S = {v1...vp} be a set in V and let H = span {v1...vp}. If H does not equal {0}, then
use the pivot columns of A How would you form a basis for the column space of A if you knew its row equivalent matrix B?
a unique set of scalars c1...cn such that x = c1b1 + ... + cnbn Let B = {b1...bn} be a basis for a vector space V. Then for each x in V, there exists
the weights c1...cn such that x = c1b1 + ... + cnbn Suppose B= {b1...bn} is a basis for a vector space V and x is in V. The coordinates of x relative to the basis B (or the B-coordinates of x) are
the vector whose entries from top to bottom are c1...cn, the weights which solve the equation x = c1b1 + ... + cnbn The coordinate vector of x relative to B or B- coordinate vector of x or [xb] =
linearly dependent If a vector space V has a basis B = {b1...bn} then, any set containing more than n vectors must be
every basis of V must consist of n vectors If a vector space V has a basis of n vectors, then
the number of vectors in a basis for V The dimension of a vector space V, dim V, is
any linearly independent set of exactly p vectors, any set of exactly p vectors that spans V Let V be a p-dimensional vector space where p is greater than or equal to 1. ________ or _________ is a basis for V
the nonzero rows of B form a basis for the row space of A as well as B If a matrix A is row equivalent to B, an echelon form, then
their row spaces are the same If two matrices A and B are row equivalent then
the dimension of the column space of A (the number of pivot columns of A, the dimension of the row space of A) The rank of a matrix A is
rankA + dimNulA = n If A is an m x n matrix, what does the rank theorem say
rankA The rank of A transpose is
vector with nonnegative entries that sum to 1 A probability vector (aka state vector) is a
square matrix whose columns are probability vectors A stochastic matrix is a
sequence of probability vectors defined by xsubk+1 = A(xsubk) where A is a stochastic matrix A Markov chain is a
x = Mx where M is a stochastic matrix A vector x is called a steady state vector if
nonzero vector x such that Ax = λx for some scalar λ An eigenvector of an n x n matrix A is a
scalar λ, there is a nontrivial solution x of Ax = λx A ______ is an eigenvalue if ______
the set of all solutions to (A-λI)x = 0 for a fixed λ An eigenspace of A corresponding to λ
entries on its main diagonal The eigenvalues of a triangular matrix are the
{v1...vr} is a linearly independent set If v1...vr are eigenvectors corresponding to distinct eigenvalues λ1...λr of an n x n matrix A, then
they have the same characteristic polynomial and hence the same eigenvalues with same multiplicities If n x n matrices A and B are similar then
det(A-λI) = 0 the characteristic equation is
det(A-λI) the characteristic polynomial is

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