M340L Exam 3 T/F

FALSE
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.
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TRUE
The columns of P C<---B are linearly independent.
FALSE
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
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
To find the eigenvalues of A, reduce A to echelon form.
FALSE
the vector must be nonzero
If Ax = λx for some scalar λ, then x is an eigenvector of A.
FALSE
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
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
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.
An elementary row operation on A does not change the determinant.
TRUE
(det A)(det B) = det AB
FALSE
-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
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
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.
FALSE
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
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
A is diagonalizable if and only if A has n eigenvalues, counting multiplicities.
FALSE
A can be diagonalizable and not invertible
If A is diagonalizable, then A is invertible.
FALSE
the eigenvectors have to linearly independent
A is diagonalizable if A has n eigenvectors.
FALSE
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
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]
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
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)
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)
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.
FALSE
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.
FALSE
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]
If the square matrix A is diagonalizable, then A is invertible
FALSE
n LI eigenvectors
An n x n matrix A is diagonalizable if and only if A has n eigenvalues, counting
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
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.
FALSE
IIc v(vector)II = IcI IIv(vector)II
For any scalar c and vector v in R^n
, IIcvII = cIIvII.
TRUE
Not every orthogonal set in R^n
is linearly independent
FALSE
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
FALSE
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
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
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
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.
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