Linear Algebra Exam 3 (Definitions)

Definitions

A basis for a vector space

A basis for a vector space V is a set S in V which spans V and is linearly independent

The dimension of a finite-dimensional vector space

The dimension of a finite dimensional vector space V is the size of any basis for V

Row, column, and null space

If A is an mxn matrix, the row space of A (row(A)) is the subspace of IR spanned by the rows of A, the column space of A (col(A)) is the subspace of IR spanned by the columns of A, and the null space of A (null(A)) is the set of X in IR such that AX=0 (interpreting X as an nx1 matrix)

The orthogonal complement of a subspace of some IR

The orthogonal complement of w, is the set of vectors orthogonal to each vector in W

The projection theorem

Suppose that W is a subspace of IR. Each v in IR can be written in a unique way as v=w+w^T, where w is in W and w’ is in W^T

Rank-nullity theorem

If A is an nxn matrix, then rank(A) + nullity(A) = m