**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**-

Thu May 28 14:15:45 PDT 1998