FALSE
it’s the C-coordinate vectors of the vectors in the basis B
it’s the C-coordinate vectors of the vectors in the basis B
The columns of the change-of-coordinates matrix P C<---B are B-coordinate vectors of the vectors in C.
TRUE
If V= R^n and C is the standard basis for V, then P C<---B is the same as the change-of-coordinates matrix P_B introduced in Section 4.4.
TRUE
The columns of P C<---B are linearly independent.
FALSE
it satisfies [X]C = P[X]B
it satisfies [X]C = P[X]B
If V = R^2, B = {b1,b2}, and C = {c1,c2}, then row reduction of [c1 c2 b1 b2] to [I P] produces a matrix P that satisfies [x]_B = P[x]_C for all x in V.
FALSE
the vector has to be nonzero
the vector has to be nonzero
If Ax = λx for some vector x, then λ is an eigenvalue of A.
TRUE
A matrix A is not invertible if and only if 0 is eigenvalue of A.
TRUE
A number c is an eigenvalue of A if and only if the equation (A -cI)x = 0 has a nontrivial solution.
TRUE
Finding an eigenvector of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy.
FALSE
Row reducing changes the eigenvectors and eigen valuse
Row reducing changes the eigenvectors and eigen valuse
To find the eigenvalues of A, reduce A to echelon form.
FALSE
the vector must be nonzero
the vector must be nonzero
If Ax = λx for some scalar λ, then x is an eigenvector of A.
FALSE
the converse is true
the converse is true
If v1 and v2 are linearly independent eigenvectors, then they correspond to distinct eigenvalues.
TRUE
A steady-state vector for a stochastic matrix is actually an eigenvector.
FALSE
triangular matrices
triangular matrices
The eigenvalues of a matrix are on its main diagonal.
TRUE
An eigenspace of A is a null space of a certain matrix.
FALSE
A is triangular
A is triangular
The determinant of A is the product of the diagonal entries in A.
FALSE
interchanging rows and multiply a row by a constant changes the determinant.
interchanging rows and multiply a row by a constant changes the determinant.
An elementary row operation on A does not change the determinant.
TRUE
(det A)(det B) = det AB
FALSE
-5 is an eigenvalue
-5 is an eigenvalue
If λ + 5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A.
FALSE
absolute value of determinant
absolute value of determinant
If A is 3 x 3, with columns a1, a2, a3, then det A equals the volume of the parallelepiped determined by a1, a2, a3.
FALSE
detA^T = det A
detA^T = det A
det A^T = (-1)detA
TRUE
The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue
of A.
of A.
FALSE
row ops may change eigenvalues
row ops may change eigenvalues
A row replacement operation on the square matrix A does not change the eigenvalues.
FALSE
D must be a diagonal matrix
D must be a diagonal matrix
A is diagonalizable if A = PDP^-1 for some matrix D and some invertible matrix P.
TRUE
If R^n has a basis of eigenvectors of A, then A is diagonalizable.
FALSE
always has n eigenvalues, counting multiplicity
always has n eigenvalues, counting multiplicity
A is diagonalizable if and only if A has n eigenvalues, counting multiplicities.
FALSE
A can be diagonalizable and not invertible
A can be diagonalizable and not invertible
If A is diagonalizable, then A is invertible.
FALSE
the eigenvectors have to linearly independent
the eigenvectors have to linearly independent
A is diagonalizable if A has n eigenvectors.
FALSE
converse is true
converse is true
If A is diagonalizable, then A has n distinct eigenvalues.
TRUE
If AP = PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A.
FALSE
these are not directly related
these are not directly related
If A is invertible, then A is diagonalizable.
TRUE
v . v = IIvII^2
TRUE
For any scalar c, u . (cv) = c(u . v)
TRUE
If the distance form u to v equals the distance from u to -v, then u and v are orthogonal.
FALSE
[1 1
0 0]
[1 1
0 0]
For a square matrix A, vectors in Col A are orthogonal to vectors in Nul A.
TRUE
If vectors v1,…,vp span a subspace W and if x is orthogonal to each vj for j = 1,…,p, then x is in W^perp
TRUE
u . v – v . u = 0
FALSE
need absolute value of c
need absolute value of c
For any scalar c, IIcvII = cIIvII
TRUE
If x is orthogonal to every vector in a subspace W, then x is in W^perp.
TRUE
If IIuII^2 + IIvII^2 = IIu + vII^2, then u and v are orthogonal.
TRUE
***(end of hw T/F)
***(end of hw T/F)
For an m x n matrix A, vectors in the null space of A are orthogonal to vectors in the row space of A.
TRUE
***(BB start)
***(BB start)
If B and C are bases for a vector space V , then the columns of the change of
coordinates matrix from B to C are linearly independent.
coordinates matrix from B to C are linearly independent.
FALSE
vector x does not equal 0 vector
vector x does not equal 0 vector
If A is an n x n matrix and Ax = λx for some scalar λ, then x is an eigenvector
of A.
of A.
FALSE
triangular matrix
triangular matrix
The eigenvalues of a square matrix are the scalars on its main diagonal.
FALSE
eg [1 0 OR could have 0 eigenvalue
0 0]
eg [1 0 OR could have 0 eigenvalue
0 0]
If the square matrix A is diagonalizable, then A is invertible
FALSE
n LI eigenvectors
n LI eigenvectors
An n x n matrix A is diagonalizable if and only if A has n eigenvalues, counting
multiplicities.
multiplicities.
TRUE
If A and P are square matrices and AP = P D, with D diagonal, then the nonzero
columns of P must be eigenvectors of A
columns of P must be eigenvectors of A
TRUE
If vectors v1,…, vp span a subspace W of R^n and if x is orthogonal to each vj
for j = 1,…, p, then x is in the orthogonal complement of W.
for j = 1,…, p, then x is in the orthogonal complement of W.
FALSE
IIc v(vector)II = IcI IIv(vector)II
IIc v(vector)II = IcI IIv(vector)II
For any scalar c and vector v in R^n
, IIcvII = cIIvII.
, IIcvII = cIIvII.
TRUE
Not every orthogonal set in R^n
is linearly independent
is linearly independent
FALSE
give by proj_w y(vector)
***(BB end)
give by proj_w y(vector)
***(BB end)
The best approximation to a vector y in R^n by elements of a subspace W of R^n
is given by the vector y – proj_w y
is given by the vector y – proj_w y
FALSE
unique
unique
The orthogonal projection y(hat) of a vector y in R^n onto a subspace W of R^n can sometimes depend on the orthogonal basis for W used to compute y(hat)
TRUE
The dimensions of the row space and the column space of A are the same, even if A is not square.
FALSE
C-coords of the vectors in B
C-coords of the vectors in B
If B and C are bases for a vector space V, then the columns of the change-of-coordinates matrix from basis B to C are the B-coordinate vectors of the vectors in C.
FALSE
[x]c = P[x]B
[x]c = P[x]B
If B = {b1,b2} and C = {c1,c2} are two baes for R^2, then row reduction of [c1 c2 b1 b2] to [I P] produces a matrix P that satisfies [x]B = P[x]C for all x in R^2.
TRUE
A number c is an eigenvalue of a square matrix A if and only if the equation (A – cI)x = 0 has a nontrivial solution.
FALSE
one eigenvalue may have 2 or more LI eigenvectors
one eigenvalue may have 2 or more LI eigenvectors
If v1 and v2 are LI eigenvectors for some matrix A, then they correspond to distinct eigenvalues
TRUE
If R^n has a basis of eigenvectors of an n x n matrix A, then A is diagonalizable.
TRUE
If A, P, D are square matrices with D diagonal that satisfy AP = PD, then the nonzero columns of P must be eigenvectors of A.
