Midterm2practice

# Midterm2practice - 2 4 5 3 7 8 b 1 1 0 1 2 2 0 2 1 c 2 1 2...

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NAME: Math 415 — Midterm 2 practice Total points: 100 . Please show your work and explain all answers. Cal- culators, computers, books and notes are not allowed. Suggestion: even if you cannot complete a problem, write out the part of the solution you know. You can get partial credit for it. 1. [20 points] Explain why for C [ - 1 , 1] (the set of continuous functions from - 1 to 1) the following is not a valid inner product. h f,g i = Z 1 - 1 f ( x ) g ( x ) x d x 1

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NAME: 2. [20 points] Given the inner product for C [0 , 1] deﬁned as h f,g i = Z 1 0 f ( x ) g ( x ) x d x use the Gram-Schmidt method to obtain an orthonormal basis for the span of the following vectors w 1 = 1; w 2 = x ; w 3 = x 2 2
NAME: 3. [20 points] Determine which of the following matrices are positive deﬁ- nite. Prove your answer. a) 1 2 3

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Unformatted text preview: 2 4 5 3 7 8 b) 1 1 0 1 2 2 0 2 1 c) 2 1 2 1 2 2 2 2 3 3 NAME: 4. [20 points] Find the vector in the Span [ v 1 , v 2 , v 3 ] that is closest to the vector z . v 1 = 1 1 1 ; v 2 = 1-1 ; v 3 = 2 1 ; z = 1 1 1 2 4 NAME: 5. [20 points] Given the matrix A below, ﬁnd the orthogonal complements of R ( A ) and R ( A T ). How are the dimensions of the orthogonal complements you found related? A = 1 0 1 3 1 2 1 1 1 1 1 0-2-1 4 1 3 7 2 5...
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## This note was uploaded on 02/10/2009 for the course MATH 415 taught by Professor Bertrandguillou during the Spring '08 term at University of Illinois at Urbana–Champaign.

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Midterm2practice - 2 4 5 3 7 8 b 1 1 0 1 2 2 0 2 1 c 2 1 2...

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