(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

you have 4 vectors:

v1=(2,1,-1), v2=(-m,-1,3), v3=(-3,2,m+1), v4=(1,2,1)

a) for which values of m do the vectors span R^3

b) the vector w=(m+1, m-1, 1). for which m is there a solution to:

x1v1 + x2v2 + x3v2 + x4v4 = w ?

2. Relevant equations

the definition of spanning - every vector in R^3 can be made of those 4 vectors.

3. The attempt at a solution

a) i made the following matrix where the first few coloums are v4,v1,v3,v4 and the last one is some vector (a,b,c):

now the only way that there isn't a solution is if there's a row with the form:Code (Text):

[1 2 -3 -m a]

[2 1 2 -1 b]

[1 -1 (m+1) 3 c]

and reduced it to:

[1 2 -3 -m a]

[0 -3 8 (2m-1) b-2a]

[0 0 (m-4) (4-m) c-b+a]

[0 0 0 0 alpha] where alpha is any scalar. since the only row that can possibly look like this is the third one where m=4 then the answer is that they span R^3 for every m except 4.

Is that right?

b) here's where i got confused, if m doesn't equal 4 then there's a solution because according to a) the vectors v1,v2,v3 and v4 span R^3 which includes w. and if m=4 then there's no solution because for w:

c-b+a = 3 (not 0)

so the answer for b should be the same as a). is that true? it doesn't seem likely that there isn't some trick or something.

Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# LA - spanning R^3

**Physics Forums | Science Articles, Homework Help, Discussion**