Part 1 - Solutions of systems of linear equations Observation Let m = 10. Choose randomly n column vectors y1, y2, . . . , yn in R m in following cases: Case I. m = n; Case II. m > n; Case III. m < n. For each case, generate y1, y2, . . . , yn at least five times and monitor the number of times, the set {y1, y2, . . . , yn} is a linearly independent set of vectors and the number of times it is a linearly dependent set. 1. Now let b be a random vector of size m. Using the above observation, justify a typical solution set of Ax = b in each of the cases I, II and III where A is an m n random matrix. (In this part, m and n are both integers and both are greater than 10 or larger). 2. Now give examples of exceptions for each of the above cases (but for this part, you can use integers m and n that need only be larger than 2, so 3 or larger)

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