Linear Combinations: Solving for 4th Vector with 3 Vectors

[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.

 

[micromass]

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

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

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

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

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