Tracker Posted April 30, 2008 Share Posted April 30, 2008 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. Link to comment Share on other sites More sharing options...
timo Posted April 30, 2008 Share Posted April 30, 2008 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. Link to comment Share on other sites More sharing options...
Tracker Posted April 30, 2008 Author Share Posted April 30, 2008 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. Link to comment Share on other sites More sharing options...
timo Posted April 30, 2008 Share Posted April 30, 2008 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? Link to comment Share on other sites More sharing options...
cjohnso0 Posted April 30, 2008 Share Posted April 30, 2008 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. Link to comment Share on other sites More sharing options...
Country Boy Posted May 7, 2008 Share Posted May 7, 2008 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. Link to comment Share on other sites More sharing options...
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