# Math 313 Final

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- Texas
- Texas State University-San Marcos
- Math 313 Final

**Created:**2014-07-23

**Last Modified:**2016-03-08

^{2}+ 1 then A is invertible

^{2}is singular.

^{-1}AP is diagonal

^{-1}and B

^{-1}are similar

^{-1}is diagonalizable

^{T}is diagonalizable

^{n}consisting of eigenvectors of an n x n matrix A, then A is diagonalizable

^{2}is an example of a weighted inner product.

^{2}<u, v>

^{2}=0, then v=0

^{n}

^{3}has an orthonormal basis with respect to the Euclidean inner product.

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

^{T}A is a square matrix

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

^{T}A is invertible.

^{T }Ax=A

^{T}B is also consistent.

^{T}Ax=A

^{T}b is also inconsistent.

^{m}, then Ax=b has a unique least square solution.

^{T}A=I

^{2}is orthogonal and (detA)

^{2 }=1

^{T}and A

^{T}A are orthogonally diagonalizable

_{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}

^{-1}is orthogonally diagonalizable

^{n }consisting of eigenvectors of A

_{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.

_{0}is a nonzero vector in V, then the formula T(v)=v

_{0}+v defines a linear operator on V.

_{6}→M

_{22}is a linear transformation then the nullity of T is 3

_{22}→R defined by T(A)=det A is a linear transformation

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