Linear Algebra - created from Mind Map

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Vector Space

V is a vector space over a field F if there exists:

- Addition operator, results in sum

- Scalar multiplication, results in product

If F is the real numbers (R)

=> Real Vector Space

If F is the complex numbers (C)

=> Complex Vector Space

Linear Combination

The summation of the scalar products of a set S of n vectors (from vector space V over F) and a set of scalars P.

span/generator

The subspace spanned/generated by S is the set L of all linear combinations in S.

Linear dependence

Linear Dependence: when the summation of the scalar products of vectors in V and scalars in F (not all scalars = 0), equals 0.

Also, a set S of vectors in V is linearly dependent iff one vector in S is a linear combination of the others.

S is linearly dependent if one vector in S is in the subspace spanned/generated by the remaining vectors in S

Linear Independence:

When this summation does not equal zero

Linear Span

The set L that span/generates set of n vectors S has a LINEAR SPAN, P, if there is a set smaller than L that is linearly independent. Then there exists a linearly ind subset of P whose linear span also equals L.

Basis

A subset of set S, which spans the finite dimensional vector space V; this forms a BASIS for V

Once a particular basis is chosen => each vector has a unique expansion in terms of this basis