# Homework Help: Linear Combinations

1. Jan 22, 2010

### Precursor

The problem statement, all variables and given/known data
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?

Last edited: Jan 22, 2010
2. Jan 22, 2010

### 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...

Last edited: Jan 22, 2010
3. Jan 22, 2010

### Char. Limit

Do you even need that middle vector...?

4. Jan 22, 2010

### vela

Staff Emeritus
Looks right to me.

5. Jan 22, 2010

### vela

Staff Emeritus
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. Jan 22, 2010

### 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?

7. Jan 22, 2010

### Staff: Mentor

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

8. Jan 22, 2010

### 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.