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.
Best Answer
Ok so the span of a bunch of vectors is intuitively "the vectors that you can make with these ones". By make I mean by using only the operations of addition/subtraction/scalar multiplication. The span makes a vector space.
So in your question the fact that $w = 4x + y - 3z$ is telling you that "$w$ can be made from $x,y,z$". Formally this means $w\in\text{span}\{x,y,z\}$.
But the next fact, that $z = 4x + y$ is telling you that actually "$z$ can be made from $x,y$", i.e. $z\in\text{span}\{x,y\}$.
So really from this you can conclude that "$w$ can be made from $x$ and $y$ only". Indeed, replace $z$ in the definition of $w$ and you get $w = 4x + y - 3(4x+y) = -8x - 2y$, so that $w\in\text{span}\{x,y\}$.
Anyway, for your second point, lets have a think about it. The statement that $\text{span}\{w,y,z\} = \text{span}\{w,x\}$ is saying "can anything made from $w,y,x$ be made only from $w,x$ and conversely, can everything made from $w,x$ be made from $w,y$ and $z$?"
Let's answer the first of these. Well clearly we can make $w$ from $w$ and $x$. Also we have just seen that $w$ can be made from $x$ and $y$, so rearranging, we can make $y$ from $w$ and $x$. Also $z$ is made from $x$ and $y$ and $y$ is made from $w$ and $x$ so that $z$ can be made from $w$ and $x$. Thus anything made from $w,y,z$ CAN be made from $w$ and $x$.
The opposite inclusion is similar. We see that $w$ can obviously be made from $w,y$ and $z$. Also $x$ can be made from $w,y$ and $z$. So anything made from $w$ and $y$ CAN be made from $w,y$ and $z$.
Hope this helps.