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math 304 Test1sample

# math 304 Test1sample - − 1 − 3 − 1 2 − 1 1(i Find...

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MATH 304–505 Spring 2011 Sample problems for Test 1 Any problem may be altered or replaced by a different one! Problem 1 (15 pts.) Find a quadratic polynomial p ( x ) such that p (1) = 1, p (2) = 3, and p (3) = 7. Problem 2 (25 pts.) Let A = 1 2 4 1 2 3 2 0 2 0 1 1 2 0 0 1 . (i) Evaluate the determinant of the matrix A . (ii) Find the inverse matrix A - 1 . Problem 3 (20 pts.) Determine which of the following subsets of R 3 are subspaces. Briefly explain. (i) The set S 1 of vectors ( x,y,z ) R 3 such that xyz = 0. (ii) The set S 2 of vectors ( x,y,z ) R 3 such that x + y + z = 0. (iii) The set S 3 of vectors ( x,y,z ) R 3 such that y 2 + z 2 = 0. (iv) The set S 4 of vectors ( x,y,z ) R 3 such that y 2 z 2 = 0. Problem 4 (30 pts.) Let B = 0 1 4 1 1 1 2 1 3 0
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Unformatted text preview: − 1 − 3 − 1 2 − 1 1 . (i) Find the rank and the nullity of the matrix B . (ii) Find a basis for the row space of B , then extend this basis to a basis for R 4 . (iii) Find a basis for the nullspace of B . Bonus Problem 5 (15 pts.) Show that the functions f 1 ( x ) = x , f 2 ( x ) = xe x , and f 3 ( x ) = e-x are linearly independent in the vector space C ∞ ( R ). Bonus Problem 6 (15 pts.) Let V be a ²nite-dimensional vector space and V be a proper subspace of V (where proper means that V n = V ). Prove that dim V < dim V ....
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