Æ Å not every column in E is a pivot column (so EB œ! has at least one free variable) Æ Å E has linearly dependent columns

Transcription

1 is linearly independent if ÞÞÞ œ! has only the trivial solution EB œ! has only the trivial solution Ðwhere E œ ÞÞÞ@ ÓÑ every column in E is a pivot column E has linearly independent columns is linearly dependent if ÞÞÞ œ! has nontrivial solutions EB œ! has nontrivial solutions Ðwhere E œ ÞÞÞ@ ÓÑ not every column in E is a pivot column (so EB œ! has at least one free variable) E has linearly dependent columns To summarize: is linearly independent means that and whenever -"@" ÞÞÞ - :@: œ! then all the weights (coefficients) -,..., - must all be!. " : is linearly dependent means that it's possible to find weights -" ßÞÞÞß-: not all! so that ÞÞÞ œ! " " : :

2 Important Results about linear dependence and independence (for vectors in any 8 Ñ (in previous lecture, and in textbook) 1. Ö! is linearly dependent, but is linearly independent Á! " " 2. If one ß ÞÞÞß@ is!, then Ö@ ß is linearly dependent " : " : 3. Ö@ " is linearly dependent if and only if one of the vectors is a scalar multiple of the other 4. are vectors in 8 and : 8 ( more vectors in the set than there are entries in each vector that is, the matrix Ò@ "@# ÞÞÞ@ : Ó is wider than it is tall ) then Ö@ " must be linearly dependent 5. Ö@ " is linearly dependent if and only if at least one of the vectors is a linear combination of the others. 6. If Ö@ " is linearly dependent Á!, then (at least) one of the vectors must be a linear combination of the preceding vectors in the set.

3 Is the set W linearly dependent or independent? ( Sometimes, other explanations are possible too! ) 1. W œ Ö/ ß / /, / where " #ß $ Ô " Ô! Ô! Ô!! "!! /" œ ß /# œ, / $ œ ß / œ in!! "! Õ! Õ! Õ! Õ" Ô "!!!! "!! Linearly independent: by inspection, or because B œ! has only the!! "! Õ!!! " trivial solution. Ô ' Ô # 2. W œ $, " Ÿ Õ" Õ$ Linearly Independent: neither vector is a multiple of the other Ô! Ô! Ô! 3. W œ! ß " ß! Ÿ Õ! Õ! Õ" Linearly dependent because! is in the set Ô " Ô! Ô! Ô! 4. W œ! ß " ß!,! Ÿ Õ! Õ! Õ# Õ" Linearly dependent: 4 vectors in (See Important Fact 4), above) $ must be linearly dependent Ô # Ô $ Ô!! # # 5. W œ,,!! & Ÿ Õ! Õ! Õ! LInearly Á! is not a multiple is not a linear combination (See Important Fact 6), above)

4 In example 5, note that W œ set of pivot columns from an echelon matrix Ô # $! "!! # # "!!! & " Õ!!!!! Å Å Å The set of pivot columns from a matrix in an echelon form is always linearly independent (why?)

5 $ X À Ä Ô B" Ô B" )B# B$ #B B # XÐBÑ œ XÐÑ œ $B" 'B# B$ #B B$ Õ ÕB $B" 'B# B$ $B We can write this formula as a matrix vector product ) " # Ô B" Ô B# œ $ ' " # Õ B $ ' " $ $ ÕB 2 Is, œ Ô 1 in the range of X? that is, Õ 3 does XÐBÑ œ, have a solution? Í does Ô Ô B" ) " # Ô 2 B # $ ' " # œ 1 Õ$ ' " $ B$ Õ ÕB 3? Aug. matrix Solution: Ô ) " # 2 Ô " #!! " $ ' " # 1 µ ÞÞÞ µ!! "! ' Õ$ ' " $ 3 Õ!!! " # B œ " #B " # B œ B Ð œ >Ñ # # B œ ' $ B œ # Ô " Ô #! " o < B œ : œ > '! Õ # Õ! So there are infinitely many points B in for which XÐB Ñ œ,. The solution set looks like?

6 Is every, from $ in the range of X? Í Í does XÐB Ñ œ, have a solution for every possible, in? does EB œ, have a solution for every possible, in $? Yes, because E has a pivot position in every row.

7 8 7 X À Ä is called a linear transformation (function) if for all? ß@ in ß and all scalars - 8 1) œ XÐ?Ñ XÐ@Ñ and 2) XÐ-?Ñ œ -XÐ?Ñ This properties imply that also XÐ! Ñ œ XÐ!! Ñ œ XÐ! Ñ XÐ! Ñ, so XÐ! Ñ œ! Ñ œ XÐ-? Ñ Ñ œ -XÐ? Ñ.XÐ@ Ñ (the same holds for any number of term = ) / A Ñ œ Ñ. AÑ œ Ñ XÐ/ AÑ œ -XÐ? Ñ.XÐ@ Ñ /XÐAÑ