Writing a vector as a linear combination of 3 others

Question

I am having some difficulties writing the vector [1, 0 , 0] as a combination of the vectors [1,1,1], [-1 2 1] and [1,2,3].

Tried lots of times using trial and error but I can't seem to lay a hand on the appropriat scalars :(

Answer 1

You want to write the vector {1, 0, 0} as a linear combination of the vectors {1, 1, 1}, {-1, 2, 1} and {1, 2, 3}.

So what you want to do is find scalars $\alpha$, $\beta$ and $\gamma$ such that:

$$\{1, 0, 0\} = \alpha \{1, 1, 1 \} + \beta \{-1, 2, 1 \} + \gamma \{1, 2, 3 \} $$

which becomes

$$\{1,0,0\} = \{\alpha - \beta + \gamma, \alpha + 2 \beta + 2\gamma, \ \alpha + \beta + 3\gamma \}$$

Can you see this will give you a linear system of three equations with three unknowns - $\alpha, \beta, \gamma$?

Can you proceed from here?