ma265 lecture notes - Chapter 3 SUBSPACES 3.1 Introduction...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Chapter 3 SUBSPACES 3.1 Introduction Throughout this chapter, we will be studying F n , the set of all n dimensional column vectors with components from a field F . We continue our study of matrices by considering an important class of subsets of F n called subspaces . These arise naturally for example, when we solve a system of m linear ho- mogeneous equations in n unknowns. We also study the concept of linear dependence of a family of vectors. This was introduced briefly in Chapter 2, Remark 2.5.4. Other topics dis- cussed are the row space, column space and null space of a matrix over F , the dimension of a subspace, particular examples of the latter being the rank and nullity of a matrix. 3.2 Subspaces of F n DEFINITION 3.2.1 A subset S of F n is called a subspace of F n if 1. The zero vector belongs to S ; (that is, 0 S ); 2. If u S and v S , then u + v S ; ( S is said to be closed under vector addition); 3. If u S and t F , then tu S ; ( S is said to be closed under scalar multiplication). EXAMPLE 3.2.1 Let A M m n ( F ). Then the set of vectors X F n satisfying AX = 0 is a subspace of F n called the null space of A and is denoted here by N ( A ). (It is sometimes called the solution space of A .) 55 56 CHAPTER 3. SUBSPACES Proof. (1) A 0 = 0, so 0 N ( A ); (2) If X, Y N ( A ), then AX = 0 and AY = 0, so A ( X + Y ) = AX + AY = 0 + 0 = 0 and so X + Y N ( A ); (3) If X N ( A ) and t F , then A ( tX ) = t ( AX ) = t 0 = 0, so tX N ( A ). For example, if A = 1 0 0 1 , then N ( A ) = { } , the set consisting of just the zero vector. If A = 1 2 2 4 , then N ( A ) is the set of all scalar multiples of [- 2 , 1] t . EXAMPLE 3.2.2 Let X 1 ,...,X m F n . Then the set consisting of all linear combinations x 1 X 1 + + x m X m , where x 1 ,...,x m F , is a sub- space of F n . This subspace is called the subspace spanned or generated by X 1 ,...,X m and is denoted here by h X 1 ,...,X m i . We also call X 1 ,...,X m a spanning family for S = h X 1 ,...,X m i . Proof. (1) 0 = 0 X 1 + + 0 X m , so 0 h X 1 ,...,X m i ; (2) If X, Y h X 1 ,...,X m i , then X = x 1 X 1 + + x m X m and Y = y 1 X 1 + + y m X m , so X + Y = ( x 1 X 1 + + x m X m ) + ( y 1 X 1 + + y m X m ) = ( x 1 + y 1 ) X 1 + + ( x m + y m ) X m h X 1 ,...,X m i . (3) If X h X 1 ,...,X m i and t F , then X = x 1 X 1 + + x m X m tX = t ( x 1 X 1 + + x m X m ) = ( tx 1 ) X 1 + + ( tx m ) X m h X 1 ,...,X m i . For example, if A M m n ( F ), the subspace generated by the columns of A is an important subspace of F m and is called the column space of A . The column space of A is denoted here by C ( A ). Also the subspace generated by the rows of A is a subspace of F n and is called the row space of A and is denoted by R ( A )....
View Full Document

This note was uploaded on 04/23/2008 for the course MA 265 taught by Professor Bens during the Spring '08 term at Purdue University-West Lafayette.

Page1 / 16

ma265 lecture notes - Chapter 3 SUBSPACES 3.1 Introduction...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online