MATH 3A Final Theorems and Definitions

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26 Terms

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Properties of inner product

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The length (or norm) of a vector is

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Unit vector is

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The distance between two vectors is

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The distance squared is

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Definition of orthogonal

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If a vector z is orthogonal to every vector in a subspace W then…

z is said to be orthogonal to W

The set of all z that are orthogonal to W is called the orthogonal complement of W denoted by W┴

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x̅ is in W┴ if and only if

x̅ is orthogonal to every vector in W

W┴ is a subspace in IRn

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(Row A) ┴ =

Nul A

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(Col A) ┴ =

(Nul A) ┴

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Pythagorean theorem

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If S = {u1 … up} is an orthogonal set of non-zero vectors in Rn then …

S is linearly independent

S is a basis of span {u1 … up}

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Orthogonal basis definition

for a subspace W of IR is a basis of W and it is also an orthogonal set

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Orthonormal set definition

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If U is a square matrix we call it

an orthonormal matrix

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An mxn matrix U has orthonormal columns if and only if

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The eigenvalues of triangular matrix are…

the diagonal entries

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If v1…vr are eigenvectors of distinct eigenvalues, then…

{vi … vr} are linearly independent

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Similarity definition

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If A ~ B, then

their characteristic equations are the same

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