Sp07_midterm2_solutions - Midterm 2 Math 20F Lecture...

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Midterm 2, Math 20F - Lecture B (Spring 2007) 1. An n × n matrix A is called skew-symmetric if A T = - A . Specifically the set of 2 × 2 skew-symmetric matrices is given by S 2 × 2 = 0 - a a 0 : a R . Let T : L 2 × 2 S 2 × 2 be the transformation from the 2 × 2 lower triangular matrices onto the 2 × 2 skew-symmetric matrices defined as T b 0 a c = 0 - a a 0 , where a , b , c are real numbers. a) (1 point) Show that S 2 × 2 is a one-dimensional subspace of 2 × 2 matrices. Find a basis for this subspace. Solution: S 2 × 2 can be expressed as a span of a matrix, S 2 × 2 = a 0 - 1 1 0 : a R = span 0 - 1 1 0 . Therefore S 2 × 2 is a 1-dimensional subspace of the space of 2 × 2 matrices with the basis 0 - 1 1 0 . b) (2 points) Show that T is a linear transformation. Solution: We need to verify the following two properties to show that T is linear. Additivity : T b 1 0 a 1 c 1 + T b 2 0 a 2 c 2 = T b 1 + b 2 0 a 1 + a 2 c 1 + c 2 0 - a 1 a 1 0 + 0 - a 2 a 2 0 = 0 - ( a 1 + a 2 ) a 1 + a 2 0 and Multiplication with a scalar : T α b 0 a c = αT b 0 a c 0 - αa αa 0 = α 0 - a a 0
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c) (2 points) Find a basis for the kernel of T . Solution: By definition Kernel(T) = b 0 a c : T b 0 a c = 0 - a a 0 = 0 0 0 0 = b 0 0 c : b, c R = b 1 0 0 0 + c 0 0 0 1 : b, c R = span 1 0 0 0 , 0 0 0 1 . Since the matrices 1 0 0 0 , 0 0 0 1 are not multiple of each other, they are linearly independent. Therefore the set 1 0 0 0 , 0 0 0 1 is a basis for the kernel of T . 2. (Each part is 1 point) Determine whether each of the following statements is true or false. For each part circle either T (true) or F (false). You do not need to justify your answers. (i) The matrix 0 1 0 0 0 1 1 0 0 is invertible. T F True : The columns of the matrix are linearly independent.
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