Linear Algebra (MAS3105)

¡Supera tus tareas y exámenes ahora con Quizwiz!

If the sum of the multiplicities of the eigenvalues of an n x n matrix A equals n, then A is diagonalizable.

False

The eigenvalues of a matrix are equal to those of its RREF.

False

The eigenvector of a matrix are equal to those of its RREF.

False

The multiplicity of an eigenvalue equals the dimension of the corresponding eigenspace.

False

Every n x n matrix is diagonalizable.

False A=[0 1 0 0]

If A and B are n x n matrices and t is an eigenvalue both A and B, then t is an eigenvalue of AB.

False, A=B=2I then t=2, but AB= 4I, then 4=t

If A and B are n x n matrices and t is an eigenvalue both A and B, then t is an eigenvalue of A +B

False, A=B=I then t=1 But A + B= 2 I

If two matrices have the same characteristic polynomial, then they have the same eigenvectors.

False, A=[ -4 -3 B=[-3 0 3 6] 0 5]

A vector space may have more than one zero vector.

False, By theorem 7.2 the zero vector of a zero space is unique.

A diagnoal n x n matrix has n distinct eigenvalues

False, Identity matrix

In any vector space, av=0 implies that v=0

False, consider a=0 and v≠ 0

If Av=tv for some vector v, then v is the eigenvector of the matrix A.

False, for some NONZERO vector then v is the eigenvector of the matrix A.

If Av=tv for some vector v, then v is the eigenvalue of the matrix A.

False, for some NONZERO vector, then v is the eigenvalue of the matrix A.

Every diagonalizable n x n matrix has n distinct eigenvectors, then it is diagonalizable.

False, if an n x n matrix has n linearly independent eigenvectors, then it is diagonalizable

The eigenspace of an n x n matrix A corresponding to an eigenvalue t is the column space of A-tI

False, is the NULL space of A-tI

A linear transformation that is one to one is an isomophism.

False, it may fail to onto.

Every n x n matrix has a eigenvector in Rⁿ.

False, rotation of A90° has no eigenvectors in R²

An n x n matrix has n distinct eigenvalues.

False, the Identity Matrix

If t is an eigenvalue of A, the the dimension of the eigenspace corresponding to t equals the rank of A-tI

False, the NULLITY of A-tI

If, for each eigenvalue t of A, the muliplicitiy of the t equals the dimension of the corresponding eigenspace, the A is diagonalizable.

False, the characteristic polynomial must also factor as a product of linear factors.

If A is diagonalizable matrix, then there exists a unique diagonal matrix D such that A=PDP⁻1.

False, the eigenvalues of A may occur in any sequence as the diagonal entries of D.

The empty set is a subspace of every vector space.

False, the empty set contains no zero vector.

A scalar t is an eigenvalue of an n x n matrix A if and only if the equation (A-tI)x=0 has a nonzero solution.

True

Every ismorphism is linear and one to one

True

Every square matrix has a complex eigenvalue.

True

Every vector space has a zero vector.

True

If A and B are n x n matrices and v is an eigenvector both A and B, then v is an eigenvector of AB.

True

If A is a diagonalizable 6 x 6 matrix having two distinct eigenvalues of mulicities 2 and 4, then the corresponding eigenspaces of A must be 2-dimensional and 4-dimensional.

True

If P is an invertible n x n matrix and D is a diagonal n x n matirx such that A=PDP⁻1, then the columns of P form a basis for Rⁿconsisting of eigenvectors of A.

True

If P is an invertible n x n matrix and D is a diagonal n x n matirx such that A=PDP⁻1, then the eigenvalues of A are the diagonal entries of D.

True

If V is a vector space and W is a subspace of v, then W is a vector space with the same operations that are defined on V.

True

If t is an eigenvalue of a linear operator, then there are infinitely many eigenvectors of the operator that corresopnd to t.

True

If two matrices have the same charcteristic polynomial, then they have the same eigenvalues.

True

If v is an eigenvector of a matrix, then there is a unique eigenvalue of the matrix that corresponds to v.

True

Only square matrices have eigenvalues.

True

The charcteristic polynomial of an n x n matrix is a polynomial of degree n.

True

The eigenvalues of the linear operator on Rⁿ are the same as those of its standard matrix.

True

an n x n matrix A is diagonalizable if and only if there is a basis for Rⁿconsisting of eigenvectors of A.

True

If A and B are n x n matrices and v is an eigenvector both A and B, then v is an eigenvector of A +B

True.

If V is a nonzero vector space, the V contains a subspace other than itself.

True.

If W is a subspace of vector V, the the zero vector of W must equal the zero vector of V.

True.

If t is an eigenvalue of muliplicity 1 for a matrix A, the 4 is an eigenvalue of A with muliplicity 2.

True.

The set of continuous real-valued funcations defined on a closed interval [a,b] is a subspace of F([a,b]), the vector space of real-valued functions defined on [a,b].

True.


Conjuntos de estudio relacionados

Causes of the Civil War and its short and long term effects

View Set

G701-2017 Change Order, G702-1992 Application and Certificate for Payment, G703-1992 Continuation Sheet, G704-2000 Certificate of Substantial Completion

View Set

Chapter 49 - Assessment of the Musculoskeletal System

View Set