# Linear combinations?

1. May 6, 2012

### HelpMeh

Hi

If i have 3 4x1 matrices and i want to check if i can express a 4th matrix as the linear combination of the first 3.

my 3 vectors:

1 7 -2
4 10 1
2 -4 5
-3 -1 -4

can this vector be expressed a linear combination of the first 3:

54
0
-108
78

my attempt:

i made a big matrix out of them:

1 7 -2 c1 54
4 10 1 c2 0
2 -4 5 c3 -108
-3 -1 -4 c4 78

i do gaussian elimination:

1 0 1.5 -30
0 1 -.5 12

or

c1 + 1.5c2 = -30
c2 - .5c3

not sure what to do now.

2. May 7, 2012

### micromass

Staff Emeritus
So your vector is a linear combination if and only if there exists $\alpha,\beta,\gamma\in \mathbb{R}$ such that

$$\left\{ \begin{array}{l} \alpha + 7\beta -2\gamma = 54\\ 4\alpha+ 10\beta +\gamma = 0\\ 2\alpha -4\beta +5\gamma= -108\\ -3\alpha-\beta -4\gamma = 78 \end{array} \right.$$

Due to Gaussian elimination (which I did not check) you reduced this question. That is: the vector is a linear combination if and only if there exists $\alpha,\beta,\gamma\in\mathbb{R}$ such that

$$\left\{ \begin{array}{l} \alpha +1.5\gamma= -30\\ \beta -.5\gamma = 12 \end{array} \right.$$

Can you find a suitable $\alpha,\beta,\gamma$ now?? Just put $\gamma=1$ and see what the $\alpha$ and $\beta$ are.