PLEASE SHOW EACH AND EVERY STEP. THANK YOU!!! 3. [6 points Suppose that that u,

Question

PLEASE SHOW EACH AND EVERY STEP. THANK YOU!!!

3. [6 points Suppose that that u, v are vectors in R", and A is any n x n matrix. Let (u, v)v denote the usual dot product. a) Show (Au, v) - (u, ATv) b) Show that if A is an orthogonal matrix (meaning AAT AT A ), then (Au, Av) (u, v) Remark: In particular, this implies that A|u|l, so A preserves length, and implies that the distance and angle between u and v is the same as that between Au and Av. Intuitively, A is a "rigid motion" that does not affect the geometric quantities we are studying. In fact any matrix that behaves this way must be orthogonal c) Suppose that n = 2 and {u, v} form an orthonormal basis for R2. Prove that the 2 × 2 matrix vvT + uuT equals the identity matrix(). Hint for one approach: Only the identity matrix fixes every vector. First show the given matrix fixes v and u (use associativity of matrix multiplication and orthonormality), and then show it must consequently fix everything by using the fact that fu, v is a basis.

Explanation / Answer

a) (Au, v) = (Au)T v = uT AT v = uT (ATv) = (u, ATv)

b) Using part (a)
(Au, Av) = (u, ATAv)
= (u, Iv) [Since A is orthogonal that is ATA = I]
= (u, v)

c) Consider (vvT + uuT)(u) = (vvTu + uuTu)
= v(vTu) + u(uTu) [using associativity]
  = v(0) + u(1) [Since {u,v} is an orthonormal basis that is vTu = 0 and uTu = 1]
= u

Similarly (vvT + uuT)(v) = v

Using the fact that {u, v} is a basis we get that any x in R2 can be written as x = cu + dv where c and d are constants.
=> (vvT + uuT)(cu + dv) =  c (vvT + uuT) (u) + d (vvT + uuT)(v) = cu + dv

Thus, the matrix vvT + uuT fixes every element of R2.

Therefore, vvT + uuT is the identity matrix.

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