Math Help - Linear Algebra Problem Involving Span

I've attached the problem below.

The question asks me to find vectors u and v (in R^3) such that u+v is equal to a vector w with the form

|2b+4c |
| b |
| c |


In my approach, I first evaluated the span{u,v}.
If u=|u1 u2 u3|^T and v=|v1 v2 v3|^T. I don't understand why they are asking me for a specific vector u and v. Couldn't I just let either u2 and u3 or v2 and v3 be zero and u2 and u3 or v2 and v3 not picked be 1?

Since any vector in can be written as we can write as so that is a linear combination of and this means that

Thank you for the reply. I can see how that can work. In general, can't I represent W as the spans of other vectors u and v as long as the components add up to create w with the form specified?

W = span\{(2,1,1), (4,0,0)} What about the span of these two vectors?
What about W = span\{(2,0,0), (4,1,1)}?

Not quite. Consider your example:

This means that for any we have for some scalars
this means that which is not the same as the original
In other words, you can't just rearrange those two vectors.


In general if you have a problem like this you want to represent the general case as a linear combinations of some vectors (thus the span). What you would do is separate the coefficients from each other (for example in your original question: ) then factor out the coefficients (again in your original question ) to obtain a linear combination of some vectors. The span of those vectors will be equal to W. You can also use any scalar multiple of those vectors (for example and you can keep going with any scalar multiple).

Does that make sense?

Yes it does. We're essentially finding the span of u and v that makes up the column vectors (2,1,0) and (4,0,1) of a coefficient matrix that is associated with a transformation whose image is of the form {(2b+4c),(b,c)}. In other words, the image of our transformation which has the general result {(2b+4c),(b,c)}, has span(u,v).