3. Consider the inner product (p, q) = So p(x)q(x) dx on P3. Let U = span{1,x, x²}. (a) Find an orthogonal basis for U. (b) Write x³ as a sum of a vector in U and a vector in U-. (c) Find a basis for U-.

The coordinates for the vector v = (1,2,4) relative to the standard basis B = {(1,0,0), (0, 1,0), (0,0,1)} is [v] B = (1,2,4). The coordinates for v relative to the nonstandard basis B' = {(1, 0, -1), (2,1,-1), (-2, 1,4)} is [v]B¹ = (9,−1,3). Find the transition matrix P-1 from B to B' and show that this does change the coordinate of v relative to B to the coordinate of v relative to B'.

2. The vector x = H [13] vector [2]B = [8] A 1 B 2 3 D0 in R2 relative to the orthogonal basis B What is the value of a + b? {0-[-]} has coordinate

Let w,=(1,0,0,1) and w=(-1,0,0,1). Let W=(w | w w, = w-w, = 0}, where w-w, denotes the dot product of w, & w. (a) Show that W is a vector space (over the field R). (b) Find the basis of W. (c) Find the dimension of W. (d) Is the answer to part (c) unique?