Numerical experimcnts in software celli/WI be used to verify that a set V wilh tWO operations Ell and 0 is a veclOr space or a sub!;pace. Such a verification must bedone "abstractly" to take into account all possibilities for elements of V . Howcver. numerical experiments can yield counterexamples which show that V is not a vector ace or not a subspace. Use your software to verify that each of the following is 110' a subspace of M n. with the usual operations of addition of matrices and scalar multiplication:(a ) The SCt of symmetric matrices wi,h the ( I. I) entryequ:ll to 3(b) The scI of mal ri ces whose first column is [0 I ] T(c) The scI of matrices [: ~] such that lid - be I- 0

Step 1 of 5) A physical therapist wishes to learn whether an exercise program increases flexibility. She measures the flexibility (in inches) of 12 randomly selected subjects both before and after an intensive 8-week training program and obtains the data shown. Is the median flexibility before the exercise program less than the median flexibility after the exercise program Use the a = 0.05 level of significance. In Figure 11, we show the t-distribution for the sample sizes n = 5 and n = 15, along with the standard normal density curve. Determine t-Values Recall that the notation za is used to represent the z-score whose area under the normal curve to the right of za is a. Similarly let ta represent the t-value whose area under the t-distribution to the right of ta is a. See Figure 12. The shape of the t-distribution depends on the sample size, n. Therefore, the value of ta depends not only on a, but also on the degrees of freedom, n - 1. In Table VII in Appendix A, the far left column gives the degrees of freedom. The top row represents the area under the t-distribution to the right of some t-value.