Linear Algebra (MAS3105)
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.
