Linear Algebra: Vector equation, find the missing integers.

Question

I am stuck at the following math problem:

I am asked to find "r" and "k", which are integers, so that the following equation will hold true, and additionally I have to find "x", "y", "z".

The equation:

r(3, 7, 0, 5) - k(7, 1, 3, 0) = (0, x, y, z)

The answers are: k=3; r = 7;

This is about vectors. And in my textbooks' lectures this case wasn't mentioned and I am stuck on this.

Any help would be greatly appreciated!

Thanks!

Answer 1

All you need is to make the first component of the resultant vector, null. Indeed you have in the right hand side:

$$(0, x, y, z)$$

So you don't care what $x, y, z$ are. You only care to solve

$$3r - 7k = 0$$

And the easiest choice is then

$$r = 7 ~~~~~~~ k = 3$$

The other components won't be zero, so you will get a tern of parameters $x, y, z$ which will be nonzero (or anyway, arbitrary).

What you need, again, is the first component to be zero.

Then, once you understood $r = 7$ and $k = 3$ you will solve the remaining expressions:

$$7(3, 7, 0, 5) - 3(7, 1, 3, 0) = (21-21,\ 49 - 3,\ 0 - 9,\ 35 - 0)$$

so

$$(0, 46, -9, 35)$$