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- Math 313 Final

mel h.

if A is a square matrix and Ax = λx for some nonzero scalar λ then x is an eigenvector of A

True

if λ is an eigenvalue of a matrix A then linear system (λI - A)x = 0 as only the trivial solution

false

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if the characteristic polynomial p(λ) of a matrix A is p(λ) = λ^{2} + 1 then A is invertible

True, det ≠ 0

If λ is an eigenvalue of a Matrix A, then the eigenspace of A corresponding λ is the set of eigenvectors of A corresponding to λ

False

* If 0 is an eigenvalue of a matrix A, then A^{2} is singular.

True

If the eigenvalues of a matrix A are the same as the eigenvalues of the reduced row echelon form of A.

False

If 0 is an eigenvalue of a matrix A, then the set of columns of A is linearly independent.

False

Every square matrix is similar to itself. 5.2

True

If A, B, and C are matrices for which A is similar to B and B is similar to C, then A is similar to C.

True

If A is diagonalizable,then there is a unique matrix P such that P^{-1}AP is diagonal

False

If A and B are similar invertible matrices, then A^{-1} and B^{-1} are similar

True

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If A is diagonalizable and invertible, then A^{-1} is diagonalizable

True

If A is diagonalizable,then A^{T} is diagonalizable

True

If there is a basis for R^{n} consisting of eigenvectors of an n x n matrix A, then A is diagonalizable

True

If every eigenvalue of a matrix A has algebraic multiplicity 1, then A is diagonalizable.

True

The dot product on R^{2} is an example of a weighted inner product.

True

The inner product of two vectors cannot be a negative real number.

False

<u, v+w> = <v, u> + <w, u>.

True

<ku, kv> = k^{2}<u, v>

True

if <u, v> = 0, then u=0 or v=0

False

If ΙΙvΙΙ^{2}=0, then v=0

True

If A is an n x n matrix, then <u, v>=Au ⋅ Av defines an inner product on R^{n}

False

If u is orthogonal to every vector of a subspace W, then u = 0

False

If us is a vector in both W and W⊥, then u =0

True

If u and v are vectors in W⊥ then u+v is in W⊥

True

If u is a vector in W⊥ and k is a real number, then ku is in W⊥

True

If u and v are orthogonal, then Ι<u, v,>Ι=ΙΙuΙΙ ΙΙvΙΙ.

False

If u and v are orthogonal, then ΙΙu+vΙΙ=ΙΙuΙΙ+ΙΙvΙΙ

False

Every linearly independent set of vectors in an inner product space is orthogonal.

False

Every orthogonal set of vectors in an inner products space is linearly independent.

True

Every nontrivial subspace of R^{3} has an orthonormal basis with respect to the Euclidean inner product.

True

Every nonzero finite-dimensional inner product space has an orthonormal basis

True

proj_{w} x is orthogonal to every vector of W

False

If A is an n x n matrix with nonzero determinant, then A has a QR-decomposition

True

*If A is an m x n matrix then A^{T}A is a square matrix

True

*If A^{T}A is invertible, then A is invertible.

False

*If A is invertible, then A^{T}A is invertible.

True

If Ax=b is a consistent linear system, then A^{T }Ax=A^{T}B is also consistent.

True

If Ax=b is an inconsistent linear system, then A^{T}Ax=A^{T}b is also inconsistent.

False

*Every linear system has a least square solution

True

*Every linear system has a unique least square solution

False

If A is an m x n matrix with linearly independent columns and b is in R^{m}, then Ax=b has a unique least square solution.

True

The matrix [1/0/0; 0/1/0] is orthogonal

False

the matrix [1/2; -2/1] is orthogonal

False

An m x n matrix A is orthogonal if A^{T}A=I

False

A square matrix whose columns form an orthogonal set is orthogonal

False

Every orthogonal matrix is invertible

True

If A is an orthogonal matrix, then A^{2} is orthogonal and (detA)^{2 }=1

True

Every eigenvalue of an orthogonal matrix has absolute value 1

True

If A is a square matrix and ΙΙAuΙΙ=1 for all unit vectors u, then A is orthogonal

True

If A is a square matrix, then AA^{T} and A^{T}A are orthogonally diagonalizable

True

If v_{1} and v_{2} are eigenvectors from distinct eigenspaces of a symmetric matrix then ΙΙv_{1}+v_{2}ΙΙ^{2}= ΙΙv_{1}ΙΙ^{2}+ΙΙv_{2}ΙΙ^{2}

True

Every orthogonal matrix is orthogonally diagonalizable

False

If A is both invertible and orthogonally diagonalizable, then A^{-1} is orthogonally diagonalizable

True

Every eigenvalue of an orthogonal matrix has absolute value 1.

True

If A is an n x n orthogonally diagonalizable matrix, then there exists an orthonormal basis for R^{n }consisting of eigenvectors of A

False

If A is orthogonally diagonalizable,then A has real eigenvalues

True

If T(c_{1}v_{1}+c_{2}v_{2})=c_{1}T(v_{1})+c_{2}T(v_{2}) for all vectors v_{1} and v_{2} in V and all scalars c_{1} and c_{2}, then T is a linear transformation.

True

If v is a nonzero vector in V, then there is exactly one linear transformation T:V→W such that T(-v) = -T(v)

False

There is exactly one linear transformation T; V→W for which T(u+v)=T(u-v) for all vectors u and v in V

True

If v_{0} is a nonzero vector in V, then the formula T(v)=v_{0}+v defines a linear operator on V.

False

The kernel of a linear transformations a vector space

True

The range of a linear transformations a vector space.

True

If T:P_{6}→M_{22} is a linear transformation then the nullity of T is 3

False

The function t: M_{22}→R defined by T(A)=det A is a linear transformation

False

The linear transformationT:M→M defined by T(A)=[1/2;3/6]A has rank 1

True

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