The DIMENSION of a VECTOR SPACE
 Basis and Dimension:
 Assume we have a vector space V.
 If V has basis
,
then
any subset of V with > n vectors must be dependent.
 If V has an n vector basis then
any other basis must have exactly n vectors.
 A finitedimensional vector space V is spanned by a finite set,
with the dimension of V, dim(V), the size n of any basis for V.
 If V is not spanned by a finite set then V is
infinitedimensional.
 Subspace Dimensions:
 A subspace H of V has a basis with
.
 If
,
some subset S of H is basis for H.
 Assume an n element subset S of V and
.
 1.
 If S is linearly independent, then S is basis for V.
 2.
 If S spans V, then S is basis for V.

dim(Nul(A)) = number of free variables for
.

dim(Col(A)) = number of pivot columns for A.
The RANK of a MATRIX
Assume A is an
matrix.
 The Row Space for A
 Row(A) is set of all linear combinations of rows of A.
 If ,
then A and B have the same row spaces.
 If ,
and B is in echelon form, then
the nonzero rows of B form a basis for row(A).
 The Rank Theorem
 Definition:
Rank(A) = dim(Col(A)).
 The Rank Theorem:
dim(Col(A)) = dim(Row(A));
rank(A) + dim(Nul(A)) = n.
 Row(A) = Col(A^{T}).
 The nullity of A is
dim(Nul(A)).
 The four fundamental subspaces for A:
Row(A), Col(A), Nul(A) and Nul(A^{T}).
 Invertible Matrix Characterizations
 If A is
invertible, the following are equivalent.
 The columns of A form a basis for R^{n}.

Col(A) = R^{n}.

dim(Col(A)) = n.

rank(A) = n.

.

dim(Nul(A)) = 0.
Alan C Genz
20000706