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Linear Combinations

  • Thread starter Precursor
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  • #1
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Homework Statement
Write the vector (1,2,3) as a linear combination of the vectors (1,0,1), (1,0,-1), and (0,1,1).

The attempt at a solution
(1,2,3) = C1(1,0,1) + C2(1,0,-1) + C3(0,1,1)

The matrix for this is:

[tex]1....1....0....1[/tex]
[tex]0....0....1....2[/tex]
[tex]1....-1....1....3[/tex]

I reduced it to the following:

[tex]1....0....0....1[/tex]
[tex]0....1....0....0[/tex]
[tex]0....0....1....2[/tex]

Therefore, (1,2,3) is a linear combination of the vectors (1,0,1), (1,0,-1), and (0,1,1). Am I right?
 
Last edited:

Answers and Replies

  • #2
benorin
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Easy check:

[tex]\mbox{Does }\left( \begin{array}{c} 1 \\ 2 \\ 3 \end{array}\right) = \textbf{1}\, \left( \begin{array}{c}1 \\ 0 \\ 1 \end{array}\right) + \, \textbf{0}\, \left( \begin{array}{c}1 \\ 0 \\ -1 \end{array}\right) + \, \textbf{2}\, \left( \begin{array}{c}0 \\ 1 \\ 1 \end{array}\right) \, ?[/tex]​


You made an arithmedic error reducing the augmented matrix, try again...
 
Last edited:
  • #3
Char. Limit
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Do you even need that middle vector...?
 
  • #4
vela
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You made an arithmetic error reducing the augmented matrix, try again...
Looks right to me.
 
  • #5
vela
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Homework Statement
Write the vector (1,2,3) as a linear combination of the vectors (1,0,1), (1,0,-1), and (0,1,1).

...

Therefore, (1,2,3) is a linear combination of the vectors (1,0,1), (1,0,-1), and (0,1,1). Am I right?
Yes, it's a linear combination of those vectors, but you should explicitly write out what that linear combination is because that's what the problem asked for.
 
  • #6
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Ok, so all I need to do is substitute in C1, C2, and C3 in front of the appropriate vectors in the original equation?
 
  • #7
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Yes. And as benorin mentioned, it's very easy to check.
 
  • #8
benorin
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One idea conveyed here is that one may use any linearly independent set of vectors to describe a space. Cartesian (sic?) coordinates use the standard basis vectors so that the (x,y,z) style coordinate (1,2,3) is a linear combination of the vectors (1,0,0), (0,1,0), and (0,0,1). Namely,

(1,2,3) = 1*(1,0,0) + 2*(0,1,0) + 3* (0,0,1)​

But, other than their linear independence, these are not special. If you have studied linear independence, deter if the 3 vectors used in the problem match this requirement. They needn't even be boring, stick-arrow vectors, polar, cylindrical, spherical coordinates also work.
 

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