LESSON 9: Dimension and Rank

Terminology:

dimension, finite-dimensional, infinite-dimensional, rank.

Objectives:

learn how to determine the dimension of a subspace, learn about the row space for a matrix, learn the relationships between the size of a matrix A, rank(A), and the dimensions of Nul(A), Col(A) and Row(A).

Reading Assignment:

Chapter 4, Sections 4.5-4.6 (pages 250-262).

Lesson Outline

Key Ideas and Discussion:

The dimension of a vector space is the number of vectors in any basis; all bases for a given vector space must have the same size. The dimension tells us the minimum number of pieces of information that we need to describe a particular vector space. The dimension of is n, but the dimension of is n+1. The dimension of a subspace of some vector space must be less than or equal to the dimension of the vector space. If the dimension of a vector space is known to be n, any set of n linearly independent vectors taken from that space is a basis.

If A is a matrix, the dimension of Col(A) is the number of pivot columns after A has been reduced to echelon form; the dimension of Nul(A) is the number of free variables. Row(A) is the space spanned by the rows of A. When A is , the rows of A are row vectors, whose transposes are in . The rank of A is the dimension of Col(A), which turns out to be the same as the dimension of Row(A). The most important relationship between the dimensions for the different subspaces associated with A is given by the Rank Theorem which says

When A is an (square) matrix, A is invertible if and only if Rank(A) = n.

Practice Problems:

4.5.5, 7, 11, 13, 19, 21 (pp. 255-256); 4.6.1, 3, 7, 11, 17, 21 (pp. 263-265).

Assignment