VECTOR SPACE

VECTOR SPACE

1. Vectors in The Plane

   • are equal if and only if
   • Vector operations
   • Scalar multiplication
   • Properties of Vector Addition & Scalar Multiplication in the Plane
   • Vectors in
     • Properties of Vector Addition & Scalar Multiplication in Rn
   • Linear Combinations of Vectors

2. Vector Space

   • 10 properties
     • in V
     • in V

3. Subspaces of Vector Space

   • 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

4. Spanning Sets & Linear Independence

   • Linear Dependence
   • Linear Independence
   • Spanning Sets

5. Basis & DImension Vector Spaces

   • Characterization
     • Spanning set for V
     • Linearly independence subset of V
   • 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
     • To find dimension of subspace W
       • Find S1
         • Check if S1 span & Linearly Independence

6. The Rank of A Matrix & Application

   • 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

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