235pracmid - Practice Midterm Problems Math 235, Fall 2011...

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Practice Midterm Problems Math 235, Fall 2011 1. a: Write the system of equations below as a matrix equation: y - z = 1 x + 2 y + 2 = - 2 - x - 2 y - z = 3 b: Solve the system of equations using row operations. 2. a: For what vectors v = a b c does the equation Ax = v have a solution if A = 1 0 1 0 - 1 2 1 - 1 3 , and x = x 1 x 2 x 3 . b: What is the rank of the matrix A ? c: What is the dimension of im( A )? What is the dimension of ker( A )? 3. What does it mean for a vector to be in the image of a matrix A . Let A be the matrix 1 2 5 - 2 0 - 2 3 - 1 1 . Is 1 2 - 1 an element of the image of A ? Why or why not? 4. a: Define what it means for a set S to be a basis of a subspace V R n . b: Let A = 1 2 3 - 1 - 1 0 1 - 1 - 1 4 3 - 5 . Give a set of vectors that span im( A ) and that are independent. 5. a: Let A be a p × q matrix, so A gives a linear map from R q to R p . Let X 1 ,X 2 R q . Assume that
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235pracmid - Practice Midterm Problems Math 235, Fall 2011...

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