00:01
Okay, so we're considering the vector space of real polynomials of degree less than or equal to 4.
00:08
So we're considering the polynomial in the following form, right? a plus b, x plus cx square, plus d x cubic plus e x power 4.
00:19
We're considering this form.
00:21
Right, so intuitively, we could consider a basis of a space, which is 1, x, x, x ,000.
00:32
X squared x cubic and x to par four so this this can be a basis of this space all the polynomial can be can be can be formed by the linear combination of this basis right so we're considering a sub considering a w which satisfied f0 equals to f0 equals to zero and f1 equal to zero equal to so because we're considering a subspace, so we need to first prove the properties of the linear, of subspace.
01:23
So basically any linear combination of such polynomials are still in this space and any multiple of such polynomials are still in this space.
01:39
So if we consider two polynomials, fx satisfied the condition and gx satisfies the condition.
01:48
So let's call a new function hx equals to fx plus gx.
01:55
So let's see if fx also satisfy this condition.
01:59
So it's obvious because h0 equals to f0 plus g0 equals to 0 and h1 equals to f1 plus g1, equals to 0.
02:10
So hx also satisfies this condition, which means the combination of two elements in this space is also in the space.
02:23
And we also need to prove any multiple of, let's call lx equals alpha fx.
02:32
And we can test if lx is also in the space.
02:36
So l0 equals alpha f0 is 0.
02:39
So l1 is alpha f1 is zero...