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This first problem is a preparation for determinants. a b Let A = . c d Let P be the parallelogram with a vertex at the origin and sides given by...
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Linear algebra, 7 problems, need detailed solution guide. prefer digital

format, but i'lll take hand written. thanks.

1. This frst problem is a preparation For determinants. Let A = ± a b c d ² . Let P be the parallelogram with a vertex at the origin and sides given by the columns oF A , as drawn on the leFt side oF the fgure: a c b d a + b c + d ` ` ` ` ' Thus the Four vertices oF P lie at ± 0 0 ² , ± a c ² , ± b d ² , and ± a + b c + d ² . Use basic geometry to fnd the area oF P . Hint: One way to proceed is to frst fnd the length oF the horizontal line segments de- noted in the fgure. (They both have the same length.) Then notice that we can cut the shaded triangle off the top oF the parallelogram and re-attach it to the bottom, to get a more standard parallelogram oF the same area , as on the right side oF the fgure. The area oF this more standard parallelogram is (base) × (height) = × d . Thus area( P ) = ‘d . 2. Consider the equation A 1 1 - 2 - 1 3 2 x ± x 1 x 2 ² = b 1 0 0 . a) Show that A x = b has no solutions. b) The least-squares solution is the next best thing. What equation do we solve to fnd it? c) ±ind the least-squares solution b x . d) ±ind the error b - A b x . 3. A physicist is measuring the electric resistance r (in Ohms) oF a wire as a Function oF temperature t (in degrees Celsius). She fnds that t r - 10 5 0 5 . 2 20 5 . 5 In other words, at temperatures t 1 = - 10 , t 2 = 0 , t 3 = 20 she fnds resistances r 1 = 5, r 2 = 5 . 2, r 3 = 5 . 5. She guesses that the dependence oF resistance on temperature should be roughly linear r ( t ) = c + dt 1
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a) Set up the matrix problem that solves for the coefficients c,d given the data. It should look like A ± c d ² = 5 5 . 2 5 . 5 for an appropriate A . b) The matrix problem does not have an exact solution, only an approximate (least- squares) solution. Find the least-squares solution for c and d . Write down the best-fit function r ( t ) = c + dt . c) What is the error between the best-fit function and the actual data, at each measured value of t ? 4. You throw a ball into the air in order to measure the gravitational acceleration g . Re- member that motion of a ball thrown upward takes the form h ( t ) = d + vt - 1 2 gt 2 , where h ( t ) denotes the height as a function of time, d is the height the ball was thrown from, v is the initial velocity, and g is the gravitational acceleration. You’re not totally sure about the initial height d or the initial velocity v (when flinging the ball into the air, it is hard to judge exactly how hard you throw it, or where it left your hand). By looking at photos, you find that t (seconds) h (meters) . 2 2 . 5 . 4 3 . 1 . 6 3 . 3 1 2 . 6 1 . 2 1 . 65 Find the best guesses for d,v, and (your real goal) g by using the following steps a) Set up a matrix equation that captures this problem, in the form A d v g = 2 . 5 3 . 1 3 . 3 2 . 6 1 . 65 . Hint: write the fit function as h ( t ) = d · 1 + v · t + g · ( - 1 2 t 2 ) . The first column of A should consist of values of ‘1’ at the five measurement times (so the first column just contains 1’s); the second column consists of ‘ t ’ evaluated at the five measurement times; and the last column consists of ‘ - 1 2 t 2 ’ evaluated at the five measurement times. b) Find the least-squares solution for d,v,g . 2
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