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day2-soln - Review Problems 2 ICME and MS&E Refresher...

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Review Problems 2 ICME and MS&E Refresher Course September 20, 2011 Linear Algebra 1. Let S = { (4 , 18 , 6) T , (8 , 0 , 12) T , (22 , 9 , 2) T , (4 , 9 , 6) T } . Show that the vec- tors in S are linearly dependent. Solution : Observe that there are four vectors in R 3 . It was stated in lec- ture that a subset of R n may have at most 3 linearly independent vectors. Thus the vectors are not linearly independent. Specifically α 1 = 0 . 5 α 2 = 0 . 25 α 3 = 0 and α 4 = 1 . 0 satisfy 0 . 5 4 18 6 + 0 . 25 8 0 12 0 . 0 22 9 2 1 . 0 4 9 6 = 0 0 0 . Thus a = { α i : 1 i 4 } is a nontrivial weighting of S that gives the zero vector, and thus the vectors are linearly dependent. 2. Let A and b be defined by A = 2 1 3 2 0 0 and b = 2 3 1 . How many x s satisfy Ax = b ? Solution : No solutions satisfy this linear system as b / ∈ R ( A ). No linear combination of the first and second column of A will have a non-zero third component. 3. Prove that A T A and A share the same nullspace. Recall that N ( A ) = { x R n | Ax = 0 } This implies that a unique least squares solution exists iff A has full col- umn rank. 1
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Solution : If x ∈ N ( A ) then by definition Ax = 0 A T Ax = 0 x ∈ N ( A T A ) In the second step, we just multiplied both sides of the equation by A T . If x ∈ N ( A T A ) then (multiplying both sides by x T ) A T Ax = 0 x T A T Ax = 0 | Ax 2 2 = 0 Ax = 0 Here we have used the property of a norm that it is only zero if the entries are zero. Thus Ax = 0 x ∈ N ( A ). This completes the proof. 4. Color perception. Human color perception is based on the responses of three different types of color light receptors, called cones. The three types of cones have different spectral response characteristics and are called L , M and S because they respond mainly to long, medium and short wave- lengths, respectively. In this problem we will divide the visible spectrum into 20 bands, and model the cones’ response as follows: L cone = 20 i =1 i p i , M cone = 20 i =1 m i p i , and S cone = 20 i =1 s i p i , where p i is the incident power in the i th wavelength band, and i , m i and s i are nonnegative constants that describe the spectral response of the different cones. The perceived color is a complex function of the three cone responses, i.e. , the vector ( L cone , M cone , S cone ), with different cone response vectors perceived as different colors. (Actual color perception is
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