Linear Algebra Test 3
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45 terms
Greek | 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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