linear algebra, unit 6

Inner product (aka dot product)

• u\cdot v=u^{T}v

• transpose is used to facilitate a dot product in Linear Algebra

Orthogonal vectors

• When two vectors are perpendicular

• u\cdot v=0

orthogonal compliment

• A vector that is orthogonal to entire subspace

What is the orthogonal compliment to Row(A)

• Row\left(A\right)\perp Nul\left(A\right)

what is Row(A)

• Row(A) is the space spanned by each row of A

Complex conjugate

• if z = i, then z̄ = -i

• the complex conjugate reverses the sign of “i”

Complex Euclidean inner product(complex dot product)

• if u is in R_n

• if v is in C_n

• u\cdot v=u_1\cdot\overline{v}_1+.\ldots+u_{n}\cdot\overline{v}_{n}

Orthogonal set

• A set where all vectors are orthogonal to each other

• All orthogonal sets are a basis since the vectors do not overlap in direction

Proof that orthogonal sets are a basis

Orthogonal Span(with proof)

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projection of y onto u(with proof)

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orthogonal decomposition theorm

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best approximation theorm

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The gram Schmidt process

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What is a least squares’ problem

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Least squares problem solution(deriviation)

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Orthonormal set

• A set of vectors that normalized and are all orthogonal