The DIMENSION of a VECTOR SPACE

Basis and Dimension:

Assume we have a vector space V.

• If V has basis $B = \{ {\bf b}_1, \dots, {\bf b}_n \}$,
  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 finite-dimensional 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 infinite-dimensional.

Subspace Dimensions:

• A subspace H of V has a basis with $dim(H) \leq dim(V)$.
• If $H \neq \{{\bf0}\}$, some subset S of H is basis for H.
• Assume an n element subset S of V and $dim(V) = n \geq 1$.

  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 $A{\bf x}= {\bf0}$.
• dim(Col(A)) = number of pivot columns for A.

The RANK of a MATRIX

Assume A is an $m \times n$ matrix.

The Row Space for A

• Row(A) is set of all linear combinations of rows of A.
• If $A \sim B$, then A and B have the same row spaces.
• If $A \sim B$, 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 $n \times n$ invertible, the following are equivalent.

• The columns of A form a basis for Rⁿ.
• Col(A) = Rⁿ.
• dim(Col(A)) = n.
• rank(A) = n.
• $Nul(A) = \{{\bf0}\}$.
• dim(Nul(A)) = 0.