MatrixRepresentations

MatrixRepresentations - Matrix Representations of Linear...

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: Matrix Representations of Linear Transformations and Changes of Coordinates 0.1 Subspaces and Bases 0.1.1 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under addition and scalar multiplication: (1) V (2) u , v V = u + v V (3) u V and k R = k u V Equivalently, V is a subspace if a u + b v V for all a,b R and u , v V . (You should try to prove that this is an equivalent statement to the first.) Example 0.1 Let V = { ( t, 3 t,- 2 t ) | t R } . Then V is a subspace of R 3 : (1) V because we can take t = 0 . (2) If u , v V , then u = ( s, 3 s,- 2 s ) and v = ( t, 3 t,- 2 t ) for some real numbers s and t . But then u + v = ( s + t, 3 s + 3 t,- 2 s- 2 t ) = ( s + t, 3( s + t ) ,- 2( s + t )) = ( t , 3 t ,- 2 t ) V where t = s + t R . (3) If u V , then u = ( t, 3 t,- 2 t ) for some t R , so if k R , then k u = ( kt, 3( kt ) ,- 2( kt )) = ( t , 3 t ,- 2 t ) V where t = kt R . Example 0.2 The unit circle S 1 in R 2 is not a subspace because it doesnt contain = (0 , 0) and because, for example, (1 , 0) and (0 , 1) lie in S but (1 , 0) + (0 , 1) = (1 , 1) does not. Similarly, (1 , 0) lies in S but 2(1 , 0) = (2 , 0) does not. A linear combination of vectors v 1 ,..., v k R n is the finite sum a 1 v 1 + + a k v k (0.1) which is a vector in R n (because R n is a subspace of itself, right?). The a i R are called the coefficients of the linear combination. If a 1 = = a k = 0, then the linear combination is said to be trivial . In particular, considering the special case of R n , the zero vector, we note that may always be represented as a linear combination of any vectors u 1 ,..., u k R n , u 1 + + 0 u k = This representation is called the trivial representation of 0 by u 1 ,..., u k . If, on the other hand, there are vectors u 1 ,..., u k R n and scalars a 1 ,...,a n R such that a 1 u 1 + + a k u k = 1 where at least one a i 6 = 0, then that linear combination is called a nontrivial representation of . Using linear combinations we can generate subspaces, as follows. If S is a nonempty subset of R n , then the span of S is given by span( S ) := { v R n | v is a linear combination of vectors in S } (0.2) The span of the empty set, , is by definition span( ) := { } (0.3) Remark 0.3 We showed in class that span( S ) is always a subspace of R n (well, we showed this for S a finite collection of vectors S = { u 1 ,..., u k } , but you should check that its true for any S ). Let V := span( S ) be the subspace of R n spanned by some S R n . Then S is said to generate or span V , and to be a generating or spanning set for V . If V is already known to be a subspace, then finding a spanning set S for V can be useful, because it is often easier to work with the smaller spanning set than with the entire subspace V , for example if we are trying to understand the behavior...
View Full Document

This note was uploaded on 02/22/2012 for the course EE 441 taught by Professor Neely during the Spring '08 term at USC.

Page1 / 16

MatrixRepresentations - Matrix Representations of Linear...

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