Midterm 2-Fall 2008-Solutions

Midterm 2-Fall 2008-Solutions - AMS 510.01 Fall 2008...

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Unformatted text preview: AMS 510.01 Fall 2008, Midterm #2 — Solutions 1. Indicate whether each of the following statements is true or false. (2 points each). (a) A one-to-one mapping can be defined from the set of natural numbers to the set of real numbers. • True/False. This question was somewhat poorly defined. A one-to-one mapping from N to R certainly exists (since N ⊂ R ). However, a one-to-one correpsondence can not be defined ( R is not countable). (b) 4 2 1 3 7 2 3 2 1 1 2 =- 4 • True. 4 2 1 3 7 2 3 2 1 1 2 = (- 1) 4 2 1 7 2 3 1 = (- 1)(1) 2 1 2 3 =- (2 · 3- 1 · 4) =- 4 (c) The expression A 1 /p , p ∈ R , p > 0, is well-defined for any matrix, A , that is diagonaliz- able. • False. The eigenvalues of A must be non-negative for A 1 /p to be well-defined. (d) An orthonormal basis can always be defined for any vector space with a defined inner product. • True. The Gram-Schimdt orthonormalization procedure can always be applied. (e) A one-to-one mapping can be defined from the set of natural numbers to the set of integers. • True. In this case, there is no ambiguity, a one-to-one correspondence between N and Z can be defined ( Z is countable). (f) The following matrix, B , is diagonalizable: B = 2 5 5- 1 5 5 2 • True. This is a real, symmetric matrix. Therefore, it is diagonalizable. (g) The mapping from R 2 → R 3 defined by F ( x, y ) = ( x + y, x- y, 2) is linear. • False. F (0 , 0) = (0 , , 2) 6 = (0 , , 0). (h) A infinite series, ∑ ∞ i =1 u i , converges if and only if the corresponding sequence { u i } con- verges to zero ( i.e. lim i →∞ u i = 0). • False. Convergence of the sequence to zero is a necessary, but not sufficient, condi- tion. For example, u i = 1 i converges to zero, but the sum is divergent. (i) The following limit is undefined: lim x →∞ cos x x • True. While the limit of cos x as x → ∞ is undefined, since the function oscillates, the value is bounded by ± 1. Since the limit of 1 x is zero, the overall limit is also zero. (j) The following function is continuous on the entire real domain ( i.e. (-∞ , ∞ )): f ( x ) = x 2- 4 x + 3 x- 1 • True. The denominator cancels out when the numerator is factored. f ( x ) = x 2- 4 x + 3 x- 1 = ( x- 1)( x- 3) x- 1 = ( x- 3) 2. For each of the following mappings on R 3 , identify whether the mapping is linear or not. Prove your statement. (10 points) • To prove that a map is linear, we must demonstrate that f ( a~x + b~ y ) = af ( ~x ) + bf ( ~ y ) for all vectors ~x and ~ y and for any scalars a and b . Note that the requirement that f ( ~ 0) = ~ is a consequence of this, where a = b = 0. To disprove linearity, it is sufficient to provide a counter example....
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Midterm 2-Fall 2008-Solutions - AMS 510.01 Fall 2008...

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