Let W, X, Y and Z be full column rank real matrices. What are the conditions (if any) on W, X, Y and Z, such that there exists a real matrix A such that A X = Z and YT A = W T ? Find all such A when the conditions are satisfied. How many linearly independent equations were there? What was the dimensionality of the left and right nullspaces? Did the rank-nullity theorem hold? The previous exercise can provide a simple way to approach this problem.

Paula makes stained-glass windows and sells them to boutique stores. if her costs total $12,000 per year plus $4 per window for the frame. how many windows must she produce to earn a profit of at least $48,000 in one year if she sells the windows for $28 each? 1. define a variable for the situation. 2. write an inequality that represents her profit. note: revenue is money coming in. cost is money going out. profit is the difference between the revenue and the cost. in other words: revenue - costs profit 3.using words, describe how many windows she must sell to have a profit of at least $48,000.