Chapter 1.1, Problem 14ES

4. Suppose that A is made up of 5 column vectors in R³, and suppose that the rank(A)=3. a. How many solutions are there to Ax=0? Justify. b. What is a geometric description for the nullspace(A)? Justify. c. Do the column vectors of A span R³? Justify. d. Is A invertible? Justify.

8.2.5 An article in Obesity Research [“Impaired Pressure Natriuresis in Obese Youths” (2003, Vol. 11, pp. 745–751)] described a study in which all meals were provided for 14 lean boys for three days followed by one stress test (with a video-game task). The average systolic blood pressure (SBP) during the test was 118.3 mm HG with a standard deviation of 9.9 mm HG. Construct a 99% one-sided upper confidence interval for mean SBP. 8.2.6 An article in Medicine and Science in Sports and Exercise [“Maximal Leg-Strength Training Improves Cycling Economy in Previously Untrained Men” (2005, Vol. 37, pp. 131–136)] studied cycling performance before and after 8 weeks of leg-strength training. Seven previously untrained males performed leg-strength training 3 days per week for 8 weeks (with four sets of five replications at 85% of one repetition maximum). Peak power during incremental cycling increased to a mean of 315 watts with a standard deviation of 16 watts. Construct a 95% confidence…

3. Suppose that A is 5 x 5 and rank(A)=4. Use this information to answer the following. a. Give a geometric description of nullspace(A). Justify. b. Is A invertible? Justify. c. Give a geometric description of the span of the column vectors of A. What space are the column vectors of A in? Justify. d. What is determinant of A? Justify.