I have been given the following problem:
Let x, y, z be non-zero vectors and suppose w = 4x + y -3z.
a) If z = 4x + y, then w = _x + _y.
b) Using the calculation in (a), mark the statements below that must be true.
Then I'm given a number of statements of the same type. I will quote only the one for purposes of this discussion and hope to be able to do the rest myself by the end of it!
(i) Span (w, y, z) = Span (w, x)
For reference, I got w = -8x - 2y and this has been marked as correct (whew!)
I'm confused by this question on a couple of fronts. Firstly, I understood a span to be the set of all linear combinations of all vectors in a given space. However, the equation given for w looks like a specific linear combination (i.e. 4 of x and 1 of y etc). Is my understanding of the definition incorrect? Is it actually any linear combination? Am I wrong about w and it's not actually a space (I see the question doesn't say it is) Have I lost the plot completely and this has nothing to do with anything?!!!
I'm also confused about how to engage with this question. Should I be taking the equations for w, y and z and summing them together to see if I get the same as the RHS? (i.e. the sum of w and x?)
I would really appreciate any light you guys can throw on this. I have a ream of questions to answer on spans and this is one of the intro ones. It's clear to me that there's something I'm not getting.