prelim2_spring2006prac - Practice for Prelim 2 MATH 294...

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Unformatted text preview: Practice for Prelim 2 MATH 294 SPRING 2006 Problems covering the same material taken from earlier Prelims are given below. The numbers in parentheses indicate the number of points, out of 100, assigned to each prob- lem. Because the problems were taken from several prelims, the numbers in parentheses do not total 100. 1. (20) GiVen the matrix 10010 01010. 01111’ 10111 determine the bases for-im(A) and ker(A) and state the dimensions of each space. 2. (15) The matrix B is obtained from A by row reduction: 1 2 —2 0 7 1 O 4 0 —3 —2 —3 1 —l —5 0 l —3 O 5 A = , B = . —3 —4 0 —2 —3 O 0 O 1 —4 3 6 —6 5 1 O 0 0 O 0 Find bases for im(A), and ker(A) and state the dimension of each of these spaces. 3. (20) Given two bases for P2: B={1, 1+1, 1+8} and C={t—t2,t2,1—t}: a.) Determine the coordinate transformation matrix PC_,B. b.) If a polynomial p has components 1, 2, -1 with respect to the basis B, what are its components with respect to the basis C? 4. (5) Find a basis for the space of functions spanned by {sin t, sin 2t,sin t cos t}. Justify your answer. 5. (10) The vectors bl = (l,O,—3), b2 2 (1,1,3), and b3 =(0,5,6) form a basis B for R3. If v has coordinates 5, 6, -1 with respect to B, find the coordinates of v with respect to the standard basis. - 6. (10) Let T: P5 —> R be given by T(p) : p(O) —p(1) for any polynomial p in P5. a.) Show that'T is linear [Suggestionz Begin by writing down the definition of a linear transformation] b.) Find a basis for the kernel of T. 7.(15) Let OOt—Ar—t b—JD—IOO Or—AOb—d Find an orthonormal basis for im(A). 8. (15) Find the equation y = [30 + [31x of the least-squares line for the data (—1, O), (O, 1), (1,2), (2, 1). ...
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