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Exam1-S09

# Exam1-S09 - Exam I Math 235 Name Instructor 1 a Write the...

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Exam I Math 235 March 11, 2009 Name: Instructor 1. a: Write the system of equations below as a matrix equation. b: Find the solutions, if any, using Gauss elimination: x + y + 3 z = 0 , 2 x + y + 4 z = 1 , 3 x + y + 5 z = 2 . 2. a: What does it mean for a vector ~v to be in the (linear) span of a set of vectors { ~ v 1 , ~ v 2 , .... , ~ v n } ? b: Let A = 1 0 1 1 1 0 0 1 1 . Write - 1 - 1 - 1 as a linear combination of the column vectors of A . 3. a: Let T : R m R n be a linear transformation. Show that the kernel of T is a subspace of R m . b: Let A = ( 1 1 1 1 ) . Find a basis of ker ( A ). 4. a: Let T : R 2 R 2 be a linear transformation. Let T 2 = T T be the transformation obtained by applying T twice, that is T 2 ( ~x ) = T ( T ( ~x )). If the matrix of T is A , what is the matrix of T 2 ? b: A linear transformation T : R 2 R 2 is zero if T ( ~x ) = ~ 0 for all ~x R 2 . Is it possible for some linear transformation that T 2 is zero but T is not ? Justify. 5. a: Compute the products AB and BA of the two matrices A, B given below, if possible.

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Exam1-S09 - Exam I Math 235 Name Instructor 1 a Write the...

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