Is the vector a linear combination?

Question

I have been working on this problem for a while now, and it has become very long and I'm not even sure it is correct. The problem is...is the vector $$\begin{bmatrix}30 & -1 & 38 & 56 & 62 \end{bmatrix}^T$$ a linear combination of $$\begin{bmatrix}1 & 7 & 1 & 9 & 4 \end{bmatrix}^T, \\ \begin{bmatrix} 5 & 6 & 3 & 2 & 8 \end{bmatrix}^T, \\ \begin{bmatrix} 9 & 2 & 3 & 5 & 2 \end{bmatrix}^T, \\ \begin{bmatrix} -2 & -5 & 4 & 7 & 9 \end{bmatrix}^T?$$ Any help on how to solve this problem would be very helpful.

Answer 1

Try to find numbers a, b, c, d not all zero, such that: [30 -1 38 56 62] =a [1 7 1 9 4]+ b [5 6 3 2 8]+ c [9 2 3 5 2]+ d [-2 -5 4 7 9]

This amounts to solving: $$ 30= a +5b +9c -2d\\ -1=7a +6b +2c -5d\\ 38= a +3b +3c +4d\\ .... $$ You will have 5 equations, 4 unknowns. If you can find such 4 numbers not all zero. I guess this would also be the intuitive way of finding if the 1st vector is a linear combination of the next 4.