here is the textbook, https://math.berkeley.edu/~yonah/files/Linear%20Al…

cover sec 1.1-1.8 and sec 2.1, the exam 13 attachmentsSlide 1 of 13

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Page 1 of 6 Print Name__________________________PID:__________________________ Last First Math 18, Midterm Exam, Summer 2021 Duration: 2:30-4:00 Show all your work; no credit will be given for unsupported answers based on the material presented in the lectures. 1. What relation must π‘Ž, 𝑏, and 𝑐 satisfy so that the following system is consistent for all values of β„Ž and π‘˜? π‘₯ π‘Žπ‘¦ β„Ž 𝑏π‘₯ 𝑐𝑦 π‘˜ Page 2 of 6 Print Name__________________________PID:__________________________ Last First 2. Let 𝐴 be an π‘š 𝑛 matrix and let 𝑏 be a vector in 𝑅 . a. If the system 𝐴π‘₯ 𝑏 has a unique solution, must the system 𝐴π‘₯ b. If the system 𝐴π‘₯ 0 has a unique solution, must the system 𝐴π‘₯ 0 have a unique solution? 𝑏 be consistent? Page 3 of 6 Print Name__________________________PID:__________________________ Last First 3. Let 𝑀 be an π‘š 𝑛 matrix. Supposed 𝑀𝑣 𝑒 and 𝑀𝑣

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