Linear Algebra
Inverse of a 2x2 matrix
(1/ad-bc)[d -b] [-c a]
If A is invertible, then elementary row operations that reduce A ^n also reduce A ^1 to Upper I^n
false
If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.
false
transpose of identity matrix is identity matrix
true
If A and B are nxn and invertible, then B^-1A^-1 is the inverse of AB
true (pay attention to order)
Finding an eigenvector of A may be difficult, but checking whether a given vector u is in fact an eigenvector is easy. Choose the correct answer below.
true, u has to be nonzero and there has to be a nontrivial solution
subspace of Rn
-zero vector in H -for each u and v in H the sum u+v is in H -for each u i H and each scalar c, the vector cu is in H
A is invertible if eigenvalue...
0 not eigenvalue of A
det(A^-1)
1/det(A)
If A is nxn invertible matrix then for each b in R^n the equation has a unique solution x=....
A^-1b
(A+B)^T
A^T+B^T
(AB^T)
B^TA^T
dimension of col A
aka rank number of pivots
what must be true for a square upper triangular matrix to be invertible
all diagonal entries must be nonzero
det(AB)
det(A)det(B)
det(A^T)
detA
. The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)r, where r is the number of row interchanges made during row reduction from A to U
false
A product of invertible n xn matrices is invertible, and the inverse of the product is the product of their inverses in the same order.
false
A subset H of set of real numbers R Superscript n is a subspace if the zero vector is in H.
false
A subspace of set of real numbers R Superscript n is any set H such that (i) the zero vector is in H, (ii) Bold u , Bold v , and Bold u plusBold v are in H, and (iii) c is a scalar and cBold u is in H.
false
AB=AC then B=C always
false
AB=BA
false
Each line in set of real numbers R Superscript n is a one-dimensional subspace of set of real numbers R Superscript n . Choose the correct answer below.
false
If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues. Choose the correct answer below.
false
The column space of a matrix A is the set of solutions of Ax= b.
false
The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of Rm
false
he dimension of Nul A is the number of variables in the equation Axequals 0. Choose the correct answer below.
false
det(A+B)=det A+det B
false
If Axequals lambdax for some vector x, then lambda is an eigenvalue of A. Choose the correct answer below.
false, must also have nontrvial solution
Can a square matrix with two identical rows be invertible?
no bc linearlly dependent
dimension of null A
number of free variables
(rA)^T
rA^T
(A^T)^T
true
A matrix A is not invertible if and only if 0 is an eigenvalue of A. Choose the correct answer below.
true
A number c is an eigenvalue of A if and only if the equation (Aminus cI)xequals0 has a nontrivial solution. Choose the correct answer below.
true
An eigenspace of A is a null space of a certain matrix. Choose the correct answer below.
true
A steady-state vector for a stochastic matrix is actually an eigenvector. Choose the correct answer below.
true
Given vectors Bold v 1 , ..., Bold v Subscript p in set of real numbers R Superscript n , the set of all linear combinations of these vectors is a subspace of set of real numbers R Superscript n .
true
If A is an nxn matric and if k is a positive integer, then A^k denotes the product of k copies of A
true
If A is invertible, then the inverse of A^-1 is A itself
true
If A is invertible, then the inverse of A^T is (A^-1)^T
true
If B equalsStartSet Bold v 1 comma ... comma Bold v Subscript p EndSet is a basis for a subspace H of set of real numbers R Superscript n , then the correspondence Bold x maps to left bracket Bold x right bracket Subscript Upper B makes H look and act the same as set of real numbers R Superscript p .
true
If Bequals StartSet Bold v 1 comma ... comma Bold v Subscript p EndSet is a basis for a subspace H and if xequals c 1 Bold v 1 plus ... plus c Subscript p Baseline Bold v Subscript p, then c 1 ,...,c Subscript p are the coordinates of x relative to the basis B.
true
If Bold v 1,..., v p are in set of real numbers R Superscript n, then Sequals Span StartSet Bold v 1 comma ... comma Bold v Subscript p Baseline EndSet is the same as the column space of the matrix Aequals left bracket Bold v 1 font size decreased by 3 times times times Bold v Subscript p Baseline right bracket.
true
If H is a p-dimensional subspace of set of real numbers R Superscript n , then a linearly independent set of p vectors in H is a basis for H. Choose the correct answer below.
true
If Upper B is a basis for a subspace H, then each vector in H can be written in only one way as a linear combination of the vectors in B .
true
If a set of p vectors spans a p-dimensional subspace H of set of real numbers R Superscript n , then these vectors form a basis of H. Choose the correct answer below.
true
If k=0 then A^k is identified with identity matrix
true
any matrix times I is the same matrix
true
