Can You Represent a Vector as a Linear Combination in Multiple Ways?

[sponsoredwalk]

Suppose we want to express the vector v = (1,-2,5) as a linear combination of the vectors
u = (1,1,1)
v = (1,2,3)
w = (2,-1,1)

We follow the method here:

$x \ \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \ + \ y \ \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix} \ + \ z \ \begin{bmatrix} 2 \\ - 1 \\ 1 \end{bmatrix} \ = \ \begin{bmatrix} 1 \\ -2 \\ 5 \end{bmatrix} \ =\ \begin{bmatrix} x \\ x \\ x \end{bmatrix} \ + \ \begin{bmatrix} y \\ 2y \\ 3y \end{bmatrix} \ + \ \begin{bmatrix} 2z \\ - z \\ z \end{bmatrix} \ = \ \begin{bmatrix} 1 \\ -2 \\ 5 \end{bmatrix}$

and continue, but I don't understand this fully.

The way I understand a 3-tuple is that (1,2,3) is
1 in the x axis, 2 in the y-axis & 3 in the z-axis.
I can't help but return to thinking this way & wanting
to write

u = (1,1,1)
v = (1,2,3)
w = (2,-1,1)

as
$x \ \begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix} \ + \ y \ \begin{bmatrix} 1 \\ 2 \\ -1 \end{bmatrix} \ + \ z \ \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix} \ = \ \begin{bmatrix} 1 \\ -2 \\ 5 \end{bmatrix} \ =\ \begin{bmatrix} x \\ x \\ 2x \end{bmatrix} \ + \ \begin{bmatrix} y \\ 2y \\ - y \end{bmatrix} \ + \ \begin{bmatrix} z \\ 3z \\ z \end{bmatrix} \ = \ \begin{bmatrix} 1 \\ -2 \\ 5 \end{bmatrix}$

Needless to say it's because I don't understand the reason why we do it
one way and not the other, I mean it doesn't make sense because in my
underdeveloped and confused understanding of linear algebra we can
transpose the vector and switch between the two ways I've done the
matrices here & none of it makes sense.
It would help a lot if anyone could clear this up with a good
explanation!

 

[ystael]

You are being confused by the fact that $$x, y, z$$ are the variable names chosen for the coefficients of the linear combination, that's all. There is no connection between the coefficients of the linear combination and the axes. Call the coefficients $$A, B, C$$ instead if it makes you feel better.

 

[sponsoredwalk]

Okay yeah that makes sense, so I can write:

au + bv + cw = a(1,1,1) + b(1,2,3,) + c(2,-1,1) =
(a + b + 2c, a + 2b - c, a + 3b + c) = (1,-2,5)
and then write it in matrix form.

Also, bu + cv + aw = b(1,1,1) + c(1,2,3,) + a(2,-1,1) =
(b + c + 2a, b + 2c - a, b + 3c + a)
is alright, right?

 

[ystael]

Right. Letters are only letters.