Projections onto Subspaces and Orthogonal Complements

Please help me with the two problems attached. In problem 8, I especially need parts a and c. I'm not sure of my approach regarding problem 12.

Linear Programming

I am having difficulty of how to solve a linear programming problem graphically. Can you do this on an Excel Spreadsheet? I have attempted to do it on my own (see below) with some trouble. Let M = number of Everett Parkas Let R = number of Colorado Parkas Max 100M + 150R s.t. 30 M + 20R < 7200 Cutting time 45 M + 15R < 7200 Sewing time 0.8 M – 0.2R > 0 % requirement I am having some trouble formulating constraints such as the % requirement constraint. Can you proceed with a systematic step-by-step fashion with formulating these types of constraints such as: M must be at least 20% of total production M>0.2 (total production) M>0.2 (M+R) M>0.2M + 0.2R 0.8M - 0.2R> 0 ... click for more

Unitary operator, invertible matrix, operator's eigenvalues/eigenvectors

Unitary operator, invertible matrix, operator's eigenvalues/eigenvectors. See attached file for full problem description.

Linear operator that has an adjoint

We are studying an inner product spaces. See attached file for full problem description.

Objective function and constraints

Develop the objective function and constraints required for the problem. Determine optimal product mix and profit contribution. Please note that the profit contribution per pound of $1.65 for the Regular Mix, $2.00 for the Premier Mix, and $2.25 for the Holiday Mix are the coefficients to use in the objective function. Respond to questions 1,2 and 3 below. Case Problem JG’s Incorporated makes 3 nut mixes for sale to grocery chains located in the Southwest. The 3 mixes, referred to as the Regular mix, the Premier mix, and the Holiday Mix, are made by mixing different percentages of 5 types of nuts. In preparation for the autumn season, JG’s Incorporated has just purchased the following... click for more