Question: Problem I.1. Let u - 23 . - 51 . and b - ipx. (1) Determine whether b is in Span(u, v). (2) If b

Problem I.1. Let u - 23 . - 51 . and b - ipx. (1) Determine whether b is in Span(u, v). (2) If b E Span{u, v}, find constants r and s such that b = ru + sv. Hint. This is straightforward: solve a properly constructed system of linear equations. Problem I.2. Let v1 = NOT CO and v3 = CO OT OO in R3 Determine whether the set { v1, v2, v3} is linearly dependent. If so, find concrete constants C1, C2 and c3 (not all zero) such that ciVI + C2V2 + C3V3 = 0. Show your work. Hint. This is straightforward: solve a properly constructed system of linear equations. 21 + 312 Problem 1.3. Define T: R? - R3 by T(x) = 21 + 5x2 for all x = =R2. 4x1 + 732 (1) Find the standard matrix of the linear transformation T N (2) Determine whether b is in the range of T. If so. find a concrete vector x E R' such that 7(x) = b. Show your work, Hint. (1) Find A such that 7(x) = Ax. What is the size of the matrix A? (2) It suffices to solve a (properly constructed) system of linear equations. Problem I.4. Let A Find A-, if it exists Hint. Apply elementary row operations to [A | I]. Show your step-by-step work

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