00:01
So we'd here like to show that b is in the span of a1 and a2.
00:05
What that means directly is that b is a linear combination of a1 and a2.
00:10
So let's write out what that means here.
00:12
Alpha times 1, 2 plus beta times 2 minus 1.
00:17
Those are a1 and a2 respectively.
00:20
I want this to be equal to b, which is colon vector 4, 3.
00:26
You can set this up as linear equations.
00:28
So alpha plus 2 beta has to equal 4, whereas 2 alpha minus beta is equal to 3.
00:38
Let's see.
00:40
If we add 2 times this second equation to the first equation, that is to say, well, we'll set this up as a matrix.
00:53
1, 2, 2 minus 1, 4, 3.
00:59
And then what i'll do is actually subtract twice the first row from the second row.
01:05
We're going to just do gaussian elimination here.
01:09
So 2 minus 2 times 1, this is going to stay the same, is 0.
01:13
Negative 1 minus 2 times 2 is negative 1 minus 4 is negative 5.
01:20
And 3 minus 8, excuse me, is also negative 5.
01:27
So we've got that...