Pretest 3

1. Prove the theorem about orthogonal complements in Rⁿ.

2. Find the linear combination of functions sin(x), sin(2x), x which is the closest possible to the function x² in C[0,1].

3. Let A be the following matrix:

[ 2	3	4 ]
[ 5	6	7 ]
[ 1	1	1 ]
[ 2	2	2 ]

Find the vector v=(x,y,z) in R³ which makes the distance from Av to (1,1,1,1) the smallest possible.

4. For which values of t do the following vectors form a basis of R⁴:

(1, t, t, t), (t, 1, t, t), (2-t, t, t, t), (3-2t, t, 1, t).

5. Prove that the vectors (2, 3, 1), (1,1,1), (3, 1, 7) form a basis in R³ and find the transformation matrix from the standard basis to this basis. What are the coordinates of the vector (3,4,1) in this basis?