9/30/2008

Dimension of V

Consider a subspace V of R^n. The number of vectors in a basis V is called the dimension of V, denoted dim(V).

Independent vectors and spanning vectors in a subspace of R^n

Consider a subspace V of R^n with dim(V) = m. Then

1. We can find at most m linearly independent vectors in V.
2. We need at least m vectors to span V
3. If m vectors in V are linearly independent, then they form a basis of V.
4. If m vectors in V span V, then they form a basis of V.

Dimension of the image

For any matrix A, dim( im A ) = rank( A )

Rank-Nullity Theorem (The fundamental theorem of linear algebra)

For any n x m matrix A, dim( ker A ) + dim ( im A) = m.

In other words, (the nullity of A) + (the rank of A) = m

Bases of R^n

The vectors v1, ..., vm in R^n form a basis of R^n if and only if the matrix [ v1 ... vm ] is invertible.

Various characterizations of invertible matrices

For an n x n matrix A, the following statements are equivalent.

1. A is invertible.
2. The linear system Ax = b has a unique solution x, for all b in R^n.
3. rref(A) = I_n.
4. rank(A) = n.
5. im(A) = R^n.
6. ker(A) = {0}.
7. The column vectors of A form a basis of R^n.
8. The column vectors of A span R^n.
9. The column vectors of A are linearly independent.

Coordinates in a subspace of R^n

Group/Class Exercises

• Section 3.3 - 5, 7, 49, 51, 52
• Section 3.4 - 4, 6, 10, 18

Reminder: Our first exam is one week from today and it will cover Chapters 1-3.