00:06
All right, so in this question, we're given a vector space, p2, which are just all degree two polynomials with real coefficients.
00:15
And we want to know if the following three vectors span p2.
00:22
So p1 is, you know, the given polynomial that we're down, negative 2x squared plus 6x plus 1, p2 is the polynomial x squared minus 2x minus 1, and p3 is 2x minus 1.
00:33
And one thing we can do, so, you know, three -vectors spanning p2 means that i can take a linear combination and get any element of p2 that i want.
00:45
So let's just let alpha, beta, and gamma be in r.
00:51
Then alpha p1 plus beta p2 plus gamma p3 is like some arbitrary linear combination.
01:00
And if you expand all this out, so alpha times p1 is alpha times negative 2x squared plus 6x plus 1, then beta times p1, gamma times p3.
01:24
And if you do all the algebra and combine you like terms, right, we get negative 2 alpha plus beta on the x squared terms.
01:33
Term.
01:34
On the x terms, we get a 6 alpha, a negative 2 beta, and a positive 2 gamma.
01:44
And then on the constant term, we get an alpha minus beta minus gamma.
01:54
And what i claim is, for example, like 5x is not in the span of p1, p2, and p3.
02:14
So i'm claiming that these vectors do not span all of p2 because here's some polynomial, right, 5x, where, you know, i've just taken my a value to be zero and my c value to be zero, my b value to be positive 5.
02:35
This thing's not in the span of these polynomials.
02:40
One way you can do that.
02:42
So here, a is zero.
02:45
And so what i'm claiming then is that we would have to have negative 2 alpha plus beta being 0...