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235FA10-SAMPLEM1 - MTH 235 Linear Algebra Sample problems...

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MTH 235 : Linear Algebra Sample problems - Midterm 1 1. (a) Argue that Z / 5 Z is a field by using addition and multiplication tables. (b) Argue that Z / 4 Z is not a field. 2. Let V be the vector space of functions f : R R . Show that W V is a subspace, where (a) W = { f | f (1) = 0 } (b) W = { f | f (3) = f (1) } (c) W = { f | f ( - x ) = - f ( x ) } 3. Prove that W 0 = { ( a 1 , a 2 , . . . , a n ) F n | a 1 + a 2 + · · · + a n = 0 } is a subspace of F n , but W 1 = { ( a 1 , a 2 , . . . , a n ) F n | a 1 + a 2 + · · · + a n = 1 } is not. 4. Fix A M n × n ( R ). Consider the subset S = { B M n × n ( R ) | AB = BA } . Show that S is a subspace. 5. If possible, find a solution of the system of equations: 3 x 1 + 2 x 2 + x 3 = 0 - 2 x 1 + x 2 - x 3 = 2 2 x 1 - x 2 + 2 x 3 = - 1 6. Are the matrices 2 0 0 10 - 1 0 π 6 3 1 1 2 - 2 0 - 1 1 3 5 row equivalent? Explain. 7. Are the matrices A = 1 0 2 2 - 1 3 4 1 8 B = 1 1 1 0 1 2 1 2 4 invertible? If they are, find A - 1 and B - 1 . Is AB invertible? If so, find ( AB ) - 1 . 1
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8. The following vectors span R 3 : u 1 = (1 , 2 , 2) u 2 = (2 , 5 , 4) u 3 = (1 , 3 , 2) u 4 = (2 , 7 , 4) u 5 = (1 , 1 , 0) (a) Pare down the set { u 1 , u 2 , u 3 , u 4 , u 5 } to form a basis, B , for R 3 . (b) Find (5 , 4 , 7) as a linear combination of the vectors in
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