Solved: Let A be an m × n matrix whose columns are

Chapter 6, Problem 21E

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Let A be an m × n matrix whose columns are linearly independent. [Careful: A need not be square.]a. Use Exercise 19 to show that ATA is an invertible matrix.b. Explain why A must have at least as many rows as columns.c. Determine the rank of AReference:Let A be an m × n matrix. Use the steps below to show that a vector x in Rn satisfies Ax = 0 if and only if ATAx = 0. This will show that Nul A = Nul ATA.a. Show that if Ax = 0, then ATAx = 0.b. Suppose ATAx = 0. Explain why xTATAx = 0, and use this to show that Ax = 0.