Finding a Basis and Dimension of Set W in R^4

[chantella28]

The set W={(a, 2a, b, 0)| a, b eR} is a subspace of R^4. Find a basis for W and find dimW

 

[Benny]

(a,2a,b,0) = a(1,2,0,0) + b(0,0,1,0).

At this point if you can't see what a possible answer is then you should review your notes.

 

[chantella28]

i've been reading through the text, but I'm doing the course through correspondence so I don't have class notes

i am guessing from that though that dimW would be 2?

 

[Benny]

Oh ok, I wasn't aware of that, sorry.

The set you have can be expressed as W = {a(1,2,0,0) + b(0,0,1,0) | a,b e R}. That's obtained just by factoring a and b, nothing special.

In this form, it is easy to see that W can be expressed as the span of the vectors (1,2,0,0) and (0,0,1,0). So any vector in W can be expressed as a linear combination of (1,2,0,0) and (0,0,1,0). So {(1,2,0,0),(0,0,1,0)} is a spanning set for W. Also, it is clear that the set is linearly independent. What can you say about a spanning set for W which is linearly independent?

Edit: You're correct about the dimension.

 

[chantella28]

ah... it forms a basis... thanks for explaining this, i understand it now... wish my textbook would just explain it like that instead of getting into a whole bunch of mathematical proof stuff expecting us to figure out how to solve it from that