Solving Linear Combinations: (1,2,3)

[Precursor]

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:

$$1...1...0...1$$
$$0...0...1...2$$
$$1...-1...1...3$$

I reduced it to the following:

$$1...0...0...1$$
$$0...1...0...0$$
$$0...0...1...2$$

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

 

[benorin]

Easy check:

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

You made an arithmedic error reducing the augmented matrix, try again...

 

[Char. Limit]

Do you even need that middle vector...?

 

[vela]

You made an arithmetic error reducing the augmented matrix, try again...

Looks right to me.

 

[vela]

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.

 

[Precursor]

Ok, so all I need to do is substitute in C1, C2, and C3 in front of the appropriate vectors in the original equation?

 

[Mark44]

Yes. And as benorin mentioned, it's very easy to check.

 

[benorin]

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.