VECTOR SPACE - created from Mind Map

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Vectors in The Plane

Vector Space

Subspaces of Vector Space

Spanning Sets & Linear Independence

Basis & DImension Vector Spaces

The Rank of A Matrix & Application

are equal if and only if

Vector operations

Scalar multiplication

Properties of Vector Addition & Scalar Multiplication in the Plane

Vectors in

Linear Combinations of Vectors

Properties of Vector Addition & Scalar Multiplication in Rn

10 properties

in V

in V

Let V a vector space, W a nonempty subset; If V and W satisfy the 10 properties, then W is called the subspace of V

Test for subspace

must be included

Linear Dependence

Linear Independence

Spanning Sets

Characterization

Finite (can count) dimensional if there is a finite subset of V

If there is no such finite subset of V, then V is called infinite dimensional

dim(W) subspace = number of matrix/vectors in basis S1

Spanning set for V

Linearly independence subset of V

To find dimension of subspace W

Find S1

Check if S1 span & Linearly Independence

If an m x n matrix A is row-equivalent to an m x n matrix B, then the row space of A is equal to the row space of B

If a matrix A is row-equivalent to a matrix B in row-echelon form, then the nonzero row vectors of B form a basis for the row space of A

Equivalence

A ~ B <-> PAQ = B for nonsingular P and Q

If A and B are two m x n matrix then they are row equivalent matrices iff row spaces of A and B are equal

The row rank and the column rank of matrix are equal

An n x n matrix A is nonsingular iff rank A = n

The linear system Ax = b has a solution iff