Please answer Question #4.

As all tutors were complaining missing reference, I am submitting the entire question set.

Please submit your solutions to these problems to be graded. All solutions should be clearly written and include full explanation/justification. 1. Consider the inner product space 3an with inner product (A, B) = tr[B*A). Determine the angle (in radians) between the matrices A_[1 0] and B_[1 I]. 0 1 1 1 2. Let A E 1'9"\" be an invertible matrix and H - H be a norm on 1'9\"\". Prove that the mapping || - HA 21EW1 )- IF defined by llelA = llAXll forx E anl is also a norm on .~nx1_ 3. Prove the polarization identity for real inner product spaces. That is, if 'v' is an inner product space over lit and x,y E V, prove that 4(x,y) = ||x +,V|l2 llx .Vll2. 4. In this problem, we will prove the maximum absolute row sum formula for ||A||m. Let A E men and let G: = maX1EFSmZF=1|ajjl. (a) Show that for any x E1?\"1 with \"Xllo-o = 1, llAXlloo