Feb 9

# Feb 9 - xv. But uEW by close under scalar mult. xvi.-uEW so...

This preview shows pages 1–4. Sign up to view the full content.

W is a subspace of a vector space V If W is a subset of V With the same ops + and scalar multiples as V Which is also a vector space under these ops. Quick Efficient test that WcV is a subspace of V i. 0ϵW where 0 is zero vector of V ii. + : , closure under in W u vϵW kuϵW iii. Closure under scalar mult: , kϵR uϵR kuϵW iv. Show WcV with i,ii,iii W is a VS i) Axiom 5 is satisfied ii) 1 is satisfied iii) 2 is satisfied v. vi. 3 is satisfied: , → + = + → u vϵV u v V u 3 is satisfied vii. Since V vs u+(v+w)=(u+v)+w so axiom 4 is satisfied viii. ix. (7) ku, vEw, kER u, v, E V k(u+v)=ku+kv (7) is satisfied x. xi. Only (6) is not obvious: xii. Show there is –uEW so that u+(-u)=0 xiii. Since uEV -uEV so that u+(-u)=0 xiv.So theoretically we could have –uEV, but –u not in W

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

View Full Document

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: xv. But uEW by close under scalar mult. xvi.-uEW so (6) xvii. xviii. xix. xx. Quick subspace test enables us to easily show that many obkects are VS easily by exploiting well-known VS: xxi.R 2 ,R 3 ,R n M mxn xxii. Ex. Det6ermine if W= xxiii. xxiv. xxv. Is a VS under usual + and scalar mult. Of vectors. xxvi. xxvii. i)0eW xxviii. 0= zero voctor in R 3 xxix. xxx. xxxi. xxxii. xxxiii. xxxiv. ii)u, vEW xxxv. xxxvi. xxxvii. xxxviii. xxxix. xl. xli. xlii. iii)ku= k xliii. xliv. xlv. xlvi. xlvii. xlviii. xlix. W is a subspace of R 3 W is a VS l. li. lii. liii. liv. Ex. W= lv. lvi. lvii. lviii. lix. Is it a vector space? i) 0eW lx. Not possible as x and y can’t equal 0 lxi. lxii. lxiii. Ex. lxiv. lxv. W= lxvi....
View Full Document

## This document was uploaded on 10/04/2011.

### Page1 / 4

Feb 9 - xv. But uEW by close under scalar mult. xvi.-uEW so...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online