Math 220 Summer Written Homework 3 Note: Any time you are asked to provide a “geometric description” you should state the dimension of the set and where it lives. Such as: a 4-dimensional space in (or living in) R6 . Zero-dimensional sets are points, 1D sets can be described as lines, and 2D sets as planes. If it’s all of a Rn you would just write that (such as “all of R3 ”). 1 2 −4 3 3 1 2 0 −5 0 5 10 −9 −7 8 . The reduced echelon form of A is 0 0 1 −2 0 . 1. Let A = 4 0 0 0 8 −9 −2 7 0 1 −2 −4 5 0 6 0 0 0 0 0 (a) Find the solution set to A⃗x = ⃗0 and write your answer in parametric vector form. (b) Find a basis for the column space of A and give a geometric description of the column space. (c) Find a basis for the null space of A and give a geometric description of the null space. (d) State the rank and nullity of A, and confirm that the rank-nullity theorem holds. 2. Suppose A is a 4 × 3 matrix such that the columns are linearly independent. (a) Give a geometric description of the columns space of A. What is the rank of A? (b) Suppose ⃗b is any vector in R4 . Is the matrix equation A⃗x = ⃗b always consistent? Use your answer in part (a) to justify your response. (c) What is the nullity of A? (d) Give a geometric description of the solution set to the matrix equation A⃗x = ⃗0. 3. Suppose A is a 5×5 matrix such that the only solution to the matrix equation A⃗x = ⃗0 is ⃗0. (a) What is the nullity of A? (b) What is the rank of A? (c) Are the columns of A linearly independent or linearly dependent? (d) Give a geometric description of the column space of A. (e) Suppose ⃗b is any vector in R5 . Is the matrix equation A⃗x = ⃗b always consistent? Justify your response with your answer to part (c). 1 ⃗ 4. Suppose A is a 5×5 matrix such the solution set to the matrix equation A⃗x = 0 consists 1 2 of vectors in the form ⃗x = t 3 for any real number t (so the solution set is a line 4 5 5 in R ). (a) What is the nullity of A? What is the rank of A? (b) Are the columns of A linearly independent or linearly dependent? (c) Give a geometric description of the column space of A. (d) Suppose ⃗b is any vector in R5 . Is the matrix equation A⃗x = ⃗b always consistent? Justify your response with your answer to part (c). 5. In this question, A is following the 2×2 matrix: 1 1 A= . 1 0 1 Define ⃗x0 = and xk = A⃗xk−1 . 0 (a) Confirm that A a b = a+b a 1 0 . (b) Compute ⃗x1 . That is, compute: ⃗x1 = A⃗x0 = 1 1 1 0 =? (c) Compute ⃗x2 . That is, compute: 1 1 previous answer ⃗x2 = A⃗x1 = =? 1 0 previous answer (d) Compute ⃗x6 . (e) Write out the first 7 terms of the Fibonacci Sequence (starting with F0 = 0 and F1 = 1). You can look up how the Fibonacci sequence works online first if you are unsure (I will also talk about this sequence briefly at the beginning of my live session). Do you see how this sequence is related to the vectors above? We will see this matrix much later in the course. We will learn some theory about matrices to allow us to solve for a closed form expression for the Fibonacci sequence! 6. At the end of each year the faculty for a graduate program examine the progress that each graduate student has made on their required thesis. Past records indicate that 30% of graduate students complete the thesis requirement (C) and 10% are dropped from the program for insufficient progress (D), never to return. The remaining students continue to work on their theses and remain current graduate students (G). 2 (a) Construct a 3 × 3 matrix A that describes the student population’s movement within the graduate program over each one year period. Label the rows and columns of A beginning with G and ending with D. (b) Currently, there are 100 graduate students in the program and no students have completed their thesis or been dropped from the program. Find an initial population vector x0 for the graduate student population. (c) Using A, predict how many students remain graduate students (G), have completed their thesis (C), and have been dropped from the program (D) one year from now. (d) Using A, predict how many students remain graduate students (G), have completed their thesis (C), and have been dropped from the program (D) two years from now. 3

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