Section 18

Vector spaces

intermediateviz

span (v_{1}, \dots, v_{k}) = {i \sum c_{i} v_{i}}

Set of all linear combinations.

intermediateviz

i \sum c_{i} v_{i} = 0 \Rightarrow c_{1} = \dots = c_{k} = 0

No vector in the set is redundant.

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dim (V) = # {vectores en una base}

A basis spans the space and is linearly independent.

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v = i \sum c_{i} b_{i} \Rightarrow [v]_{B} = (c_{1}, \dots, c_{n})^{T}

Represents a vector by coefficients relative to a basis.

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rank (A) = dim (Col (A))

Number of independent directions produced by A.

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nullity (A) = dim (ker A)

Number of degrees of freedom sent to the zero vector.

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n = rank (A) + nullity (A)

Splits the domain dimension into preserved and lost information.