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PracticeExamII

# PracticeExamII - 1 1-1 b = 3 5 Find the error || b-A x ||...

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MATH 214 Midterm II: Practice Exam November 14, 2007 1. Let P 1 be the space of all linear functions, i.e. functions of the form f ( x ) = ax + b , where a, b are real numbers. Consider the following linear transformation: T : P 1 -→ P 1 f -→ - 3 f + f. (a) Find the matrix of T with respect to the standard basis 1 , x . (b) Find the matrix of T with respect to the basis 1 - x, 1 + x . (c) Find the determinant of T .

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2. Find the QR -factorization of 2 - 2 18 2 1 0 1 2 0 .
3. Let V be the solution space of the linear system x 1 + x 2 + x 3 + x 4 = 0 x 1 + 2 x 2 + 5 x 3 + 4 x 4 = 0 . Find an orthonormal basis of V .

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4. Find the least-squares solution x * of the system A x = b ,

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Unformatted text preview: 1 1-1 , b = 3 5 . Find the error || b-A x * || . 5. (a) Use the classical adjoint of A = 1 0 1-1 1 0 2 3 , ﬁnd the inverse A-1 . (b) Use your answer in (a) to ﬁnd the solution of the liear system ± ± ± ± ± ± ± x 1 + x 3 = 5-x 1 + x 2 = 8 2 x 2 + 3 x 3 = 1 ± ± ± ± ± ± ± . 6. Use the determinant to ﬁnd out for which values of the constant k the matrix A = cos k 1-sin k 2 sin k cos k is invertible....
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PracticeExamII - 1 1-1 b = 3 5 Find the error || b-A x ||...

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