1. Vectors in R^n
Dot product: u.v = u_1v_1+...+u_nv_n. Norm: ||v|| = sqrt(v.v). Orthogonal: u.v = 0.
2. Linear Combinations and Span
Linear combination: c_1v_1+...+c_kv_k. Span: set of all linear combinations. Always a subspace.
3. Subspaces
Test: (1) Contains 0. (2) Closed under +. (3) Closed under scalar mult. Key subspaces: Col(A), Nul(A), Span{...}.
4. Linear Independence
Independent: only trivial solution to c_1v_1+...+c_kv_k = 0. Test by row reducing [v_1 ... v_k]; independent iff every column is a pivot column.
More than n vectors in R^n are always dependent. A set with the zero vector is always dependent.
5. Basis and Dimension
Basis: independent spanning set. Dimension: number of vectors in a basis.
Basis for Col(A): pivot columns of original A. Basis for Nul(A): direction vectors from parametric form of Ax=0.
6. Rank Theorem
rank(A) + nullity(A) = n (number of columns)
rank = #pivots = dim(Col(A)). nullity = #free variables = dim(Nul(A)).