The inner product can be used to define an inner product

Question

1. The inner product (..) can be used to define an inner product space for vectors in Rⁿ. The inner product defined for any inner product space must satisfy three important properties, which are listed below in the pars. verify the following properties or the inner product:

(a) (x,x) ≥0, with equality if and only if x = O_n.

(b) (x,y) = (y,x) for all x,y € Rⁿ.

(c) (ax+βy,z) =a(x,z)+β(y,z).

2. The scalar projection of u on to v defined as

and the vector projection of u onto v is

Assume that v≠ 0 and p is the vector projection of u on v. Show the following:

(a) u-p and p are orthogonal.

(b) u=p if and only if u is a scalar multiple of v.

3. Find the dimension of the vector (perhaps sub-)space spanned by these six vectors in R⁶:

4. Use the inverse to solve the following systems of equations:

(a)          2x-3y=3

                3x-4y=5

(b)          2x-3y=8

                3x-4y=11

(c)           2x-3y=0

                3x-4y=0

5. Draw a picture that graphically represents the determinant as the area of the parallelogram with two sides given by the vectors a=[3/1] and b= [1/2],like the one in class but using the given vectors.

6. Diagonalize these matrices:

(a)  C =[2/0 1/3 ]

(b)  D= [6/3 -4/-1]

7. Suppose A€R^n*n , and suppose that B=A-αl_n for some scalar a. How are the eigenvalues of A and B related?

8. Find the matrices associated with these quadratic forms:

(a) 3x²-5cy+y²

(b)2x²+3y²+z²+xy-2xz+3yz