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1. Use row reduction to find bases for the vector subspaces of R" spanned by the following vectors (a) (-3.1, 4, - 5), (1, 1, 1, 2), (-2,0,1,-3), (1, 1, - 2, 5) (b) (1, 3, 4), (4, 0, 1), 2) 2. Let A and B be two m by n matrices with row vector T'm and 81..... 8m- Prove that the row spaces of A and B are equal, i.e. - S(81 m) if and only if A and B are row equivalent. 3. Let be functions in F(R) (the vector space of functions R R). (a) Let be three real numbers. Consider the matrix A- (3(#1) fa(r2) fa(xa), Prove that if the rows of A are linearly independent then the functions are linearly independent. (b) Suppose that the functions all have first and second derivatives on some interval (a,b). In this case, for x E (a,b), let V(x) be the matrix f(x) (i(+) (1(()) W(I) (2) - fa(x) fi(()) Prove that if there exists x E (a,b) such that the rows of the matrix W(x) are linearly independent then the functions are linearly independent. (c) Show that the following sets of functions are linearly independent (1(2) 4. Use row reduction to determine whether the vector (1,1,1) belongs to the subspace of R3 generated by the vectors (1,3,4), (4,0,1), (3,1,2). 5. Determine the values of o for which the following system of equations has a solution. 3r1 - #2 + oura = 1 3x1 - a + ars - 5 6. Suppose n > m. Prove that a system of m homogeneous equations in n unknowns always has a nonzero solution.

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