Basic Linear Algebra

[Tracker]

This is an example out of Introduction to Linear Algebra by Gilbert Strang and I am trying to understand the solutions. So any help would be appreciated.

 

This is the problem: every combination of v = (1,-2,1) and w = (0,1,-1) has components that add to _____.

 

The answer is that the components of every cv + dw add to zero.

[timo]

What exactly is your problem? You don't understand what the question means? Just add the components of the vectors [math]\vec x = c\vec v + d\vec w[/math] (i.e. [math]x_1+x_2+x_3[/math] keeping c and d as variables and see that the result will equal zero - regardless of the values of c and d.

[Tracker]

keeping c and d as variables and see that the result will equal zero - regardless of the values of c and d.

 

I don't see how that is possible.

[timo]

It's basically because [math]c \cdot 0 = 0[/math] for any value of c. Have you tried what I proposed? What did you get and/or where did you get stuck?

[cjohnso0]

I didn't believe it at first, but Atheist is correct. I even double checked it with Excel.

 

If you multiply everything out, you see that zero is the only answer. Set up an equation like this:

 

[math]

c(v1 + v2 + v3) + d(w1 + w2 + w3) = 0

[/math]

 

Sub in the values for the v and w components and you'll see what Atheist is talking about.

[Country Boy]

The components of v, (1,-2,1), sum to 1-2+1= 0. The compnents of w, (0,1,-1), sum to 0+ 1- 1= 0. If C and D are any constants, then Cv+ Dw= (C, -2C+ D, C- D) and the components sum to C+ (-2C+D)+ (C- D)= C(1- 2+ 1)+ D(0+ 1- 1)= 0.