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MATH311 midterm2-ans

# MATH311 midterm2-ans - Math 311 midterm exam II answers 1...

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Math 311 midterm exam II answers 1. Since (3 , 3) = 2(1 , 2) + (1 , - 1) and the function f is linear, f (3 , 3) = 2 f (1 , 2) + f (1 , - 1) = 2(1 , 2 , 3) + (0 , - 2 , 4) = (2 , 2 , 10) . Although the argument above is sufficient, one can also calculate the whole matrix of f , as follows. If f ( x ) = a b c d e f x , then a b c d e f 1 2 = a + 2 b c + 2 d e + 2 f = 1 2 3 and a b c d e f 1 - 1 = a - b c - d e - f = 0 - 2 4 . Solving these equations, we get b = 1 / 3 , a = 1 / 3 , d = 4 / 3 , c = - 2 / 3 , f = - 1 / 3 , e = 11 / 3 . Thus f (3 , 3) = 1 / 3 1 / 3 - 2 / 3 4 / 3 11 / 3 - 1 / 3 3 3 = 2 2 10 . 2. Since x = ( x + x 2 ) - x 2 = 0(1) + 1( x + x 2 ) + ( - 1) x 2 , the coordinates are (0 , 1 , - 1) . 3. Row reduction brings the matrix 1 1 1 3 2 - 1 4 5 2 to the form 1 1 1 0 - 1 - 4 0 0 - 6 . This matrix has three lead columns, so the original three vectors are linearly independent. We could also have calculated the determinant 1 1 1 3 2 - 1 4 5 2 = 6 = 0 .

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