00:03
We're given the vector space p, the space of all polynomials of the real numbers, with an inner product of f and g, is the integral from negative 1 to positive 1 of f of t.
00:35
And then we also have our subspace w, which is the set of all polynomials of degree at most 3.
00:45
In part a, we're asked to find an orthogonal basis for w by applying the gram -schmidt algorithm to the set of vectors, polynomials, 1, t, 2, t squared, and 2 cubed.
01:14
Now, clearly these aren't orthogonal integral of t -squared, for example, from negative 1 to 1, but find an orthogonal basis.
01:39
First, i'll take one of the vectors, i'll call f1.
01:58
This will be a vector 0, or not 0, this will be the polynomial 1.
02:09
Find our next polynomial, consider a polynomial t minus the inner product of t with 1 over the inner product of 1 with itself, times times 1.
02:28
This is t minus integral from negative 1 to 1 of t d t over the integral from negative 1 to 1 of 1 d t times 1 which is t minus this is t squared over 2 from negative 1 to 1 over t from negative 1 to 1 over t from negative 1 to 1 times 1.
02:57
1.
03:03
And this is simply t minus 0 over 2 times 1, which is just t.
03:19
And so we get the other basis vector f2, which is t.
03:28
Now i want to find t squared minus the unit product of t squared with 1 over the inner product of 1 with itself, times 1 minus the inner product of t squared with t over the inner product of t itself times t this is equal to t squared minus integral from negative 1 to 1 of t squared d t over the integral from negative 1 to 1 of d t times 1 minus the integral from negative 1 to 1 of t cubed d t over the integral from negative 1 to 1 of t squared bt times t this is equal to t squared minus this is t cubed over 3 from negative 1 to 1 over t from negative 1 to 1 minus t to the 4th over 4 from negative 1 to 1 over t cubed over 3 from negative 1 to 1 times t 2 this simplifies to t squared minus, and this is two -thirds over two minus zero over two -thirds times t.
05:27
And this is simply t squared minus one -third.
05:35
And to write this for full numbers, i multiply through by three.
05:40
We get the third vector f3 of t, which is 3t squared minus 1.
05:55
And that's 3 so far.
05:57
There is one more basis vector.
06:00
Find it.
06:01
I'll take t cubed minus the inner product of t cubed with one over the inner product of one with itself, times one minus the inner product of t cubed with t over the end product of t cubed with t over the the inner product of 2 itself minus, sorry, times t, minus the inner product of t cubed with 3t squared minus 1 over the inner product of 3t squared minus 1 with itself, times 3t squared minus 1...