Mathematical formulas

Algebra

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Algebra
Vector spaces

Section 18

Vector spaces

intermedioviz

\operatorname{span}(v_1,\ldots,v_k)=\{\sum_i c_iv_i\}

Set of all linear combinations.

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Linear independence

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\sum_i c_iv_i=0\Rightarrow c_1=\cdots=c_k=0

No vector in the set is redundant.

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Basis and dimension

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\dim(V)=\#\{\text{vectores on una base}\}

A basis spans the space and is linearly independent.

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Coordinates in a basis

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v=\sum_i c_ib_i\Rightarrow[v]_\mathcal B=(c_1,\ldots,c_n)^T

Represents a vector by coefficients relative to a basis.

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\operatorname{rank}(A)=\dim(\operatorname{Col}(A))

Number of independent directions produced by A.

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\operatorname{nullity}(A)=\dim(\ker A)

Number of degrees of freedom sent to the zero vector.

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Rank–nullity theorem

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n=\operatorname{rank}(A)+\operatorname{nullity}(A)

Splits the domain dimension into preserved and lost information.

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