Proof Help Spans

  • Thread starter Phymath
  • Start date
184
0
Proof Help!! Spans!!

I was give the fallowing to prove and i would like it know if anyone thinks i did this right lol....

Let [tex] u_1, u_2, ..., u_k [/tex] be vectors in [tex] R^n [/tex] and [tex] A [/tex] be a [tex] m x n [/tex] matrix. Prove that if [tex] v [/tex] is int he span of [tex] u_1, u_2, ..., u_k [/tex] then [tex] Av [/tex] is in the span of [tex] Au_1, Au_2, ..., Au_k [/tex].

this is what i did to prove this...

If [tex] v [/tex] is in the span of the set [tex] S [/tex] which follows [tex] S = {u_1, u_2, ...,u_k}[/tex] then there is set of coeffients that make the following linear combonation of the elements of the set [tex]S[/tex] that follows the linear equation to prove that [tex]v [/tex] is in the span of [tex] S[/tex]...
[tex]c_1 u_1 + c_2 u_2 + ... + c_k u_k = v[/tex]

to show that [tex]Av[/tex] is in the span of [tex]Au_1, Au_2, ..., Au_k [/tex]
thus the fallowing linear equation must be true...[tex]c_1 Au_1 + c_2 Au_2 + ... + c_k Au_k = Av[/tex] factoring matrix [tex] A[/tex] gives [tex] A(c_1 u_1 + c_2 u_2 + ... + c_k u_k) \rightarrow Av[/tex] thus proving Av is in the span of [tex]Au_1, Au_2, ..., Au_k [/tex]

so did i make any mistakes is this crap? let me know please....

FOR ANYONE WHO'S LATEX ISN'T LOADING HERES THE RAW TYPING
I was give the fallowing to prove and i would like it know if anyone thinks i did this right lol....

Let u_1, u_2, ..., u_k be vectors in R^n and A be a m x n matrix. Prove that if v is int he span of u_1, u_2, ..., u_k then Av is in the span of Au_1, Au_2, ..., Au_k .

this is what i did to prove this...

If v is in the span of the set S which follows S = {u_1, u_2, ...,u_k} then there is set of coeffients that make the following linear combonation of the elements of the set S that follows the linear equation to prove that v is in the span of S...
c_1 u_1 + c_2 u_2 + ... + c_k u_k = v

to show that Av is in the span of Au_1, Au_2, ..., Au_k
thus the fallowing linear equation must be true...c_1 Au_1 + c_2 Au_2 + ... + c_k Au_k = Av factoring matrix A gives A(c_1 u_1 + c_2 u_2 + ... + c_k u_k) -> Av thus proving Av is in the span of Au_1, Au_2, ..., Au_k

so did i make any mistakes is this crap? let me know please....
 
Last edited:

mathwonk

Science Advisor
Homework Helper
10,733
912
just tell me briefly what you are trying to do. i cannot digest all that crap.
 

matt grime

Science Advisor
Homework Helper
9,394
3
the short answer is, yes, you've got it. (A is linear so it commutes with scalar multiplication and addition of vectors, that's all they're getting at in the question).
 

mathwonk

Science Advisor
Homework Helper
10,733
912
if v = some linear combination of the u's, then since matrix multiplication commutes with linear combinations, then Av is the same linear combination of the Au's. QED.
 

Related Threads for: Proof Help Spans

  • Posted
Replies
3
Views
6K
  • Posted
Replies
4
Views
6K
Replies
8
Views
13K
Replies
1
Views
2K
Replies
1
Views
4K
Replies
7
Views
891
  • Posted
Replies
8
Views
15K
  • Posted
Replies
2
Views
3K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top