hwB15 - v (1) = [-1 1 1 ] T , v (2) = [ 2-1 3 ] T , and s =...

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ENEE 241 02 * HOMEWORK PROBLEM 15 Due Fri 03/07 Solve by hand without using calculator matrix functions. Show all intermediate steps. Let v (1) = [5 3 - 1 1] T , v (2) = [5 - 1 7 3] T and v (3) = [1 1 - 10 0] T . Also, let s = [3 - 1 4 4] T . (i) (5 pts.) Determine the projection f of s onto the two-dimensional subspace defined by v (1) and v (2) . (ii) (2 pts.) Compute the norm of the error vector f - s . (iii) (9 pts.) Determine the projection p of s onto the three-dimensional subspace defined by v (1) , v (2) and v (3) (iv) (2 pts.) Compute the norm of the error vector p - s . (v) (2 pts.) What is the range of the 4 × 4 matrix with columns v (1) , v (2) , v (3) and v (4) = v (1) + v (2) + v (3) + s ? What is the projection of s onto this subspace, and what is the norm of the resulting error vector? ( No algebra is needed here. ) Solved Examples S 15.1 ( P 2.25 in textbook). Consider the three-dimensional vectors
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Unformatted text preview: v (1) = [-1 1 1 ] T , v (2) = [ 2-1 3 ] T , and s = [ 1 2 3 ] T . (i) Determine the projection s of s on the plane dened by v (1) and v (2) . (ii) Show that the projection of s on v (1) is the same as the projection of s on v (1) . (Is this result expected from three-dimensional geometry?) S 15.2 ( P 2.26 in textbook). Let a (1) , a (2) , a (3) and a (4) be the columns of the matrix A = 1 1 / 2 1 / 4 1 / 8 1 1 / 2 1 / 4 1 1 / 2 1 Determine the least squares approximation p = c 1 a (1) + c 2 a (2) + c 3 a (3) of a (4) based on a (1) , a (2) and a (3) . Also determine the relative (root mean square) error k p-a (4) k k a (4) k...
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