Graduate preliminary exam, Fall 2002

[3 hours, problems counted equally, R denotes the set of real numbers.]

Instructions: Work 7 of the 8 problems.

1. (a) Suppose that for each positive integer m, we have a set Sm of real numbers, a $m ,$ $S _ { m }$ real number αm, and a real function fm. Formulate the negation of this statement: $\alpha _ { m }$ $f _ { m }$ "There exists a positive integer m such that for every $x \in S _ { m } , x \geq \alpha _ { m }$ and $f _ { m } ( x ) = 0 . ^ { n }$

(b) The following statement is not valid: "For any positive integer m, if T is a set $\boldsymbol { \mathcal { T } }$ of positive integers such that (1) m  T and $( 2 ) \ n \in T$ implies $n + 1 \in T$ , then $T = \{ \mathrm { p o s i t i v e }$ integers n $: n \geq m \}$ ." Explain and correct the flaw.

2. For a function $f : \mathbb { R } \to \mathbb { R } .$ i, give the definition of continuity of f at a point a. For functions f and g from R to R, prove that if f is continuous at a and g is continuous at $b = f ( a )$ , then $g \circ f$ is continuous at a. [The composition $g \circ f$ is defined by $( g \circ f ) ( x ) = g ( f ( x ) )$ for ${ \mathfrak { x } } \in \mathbb { R } . ]$

3. Define what it means for a series $\textstyle \sum _ { n = 1 } ^ { \infty } a _ { n }$ (of real numbers $a _ { n } )$ to converge to a real number S. Prove that the series $\scriptstyle \sum _ { n = 1 } ^ { \infty } 1 / 1 0 ^ { n }$ converges.

4. Determine the $2 \times 2$ matrix A such that $A \left[ { \frac { 1 } { 2 } } \right] = { \left[ \begin{array} { l } { 5 } \\ { - 6 } \end{array} \right] }$ and $A \left[ { \begin{array} { c } { 0 } \\ { 1 } \end{array} } \right] = \left[ { \begin{array} { c } { 1 } \\ { - 1 } \end{array} } \right]$ Prove that the matrix A is not diagonalizable.

5. Let $v _ { 1 } , . . . , v _ { k }$ be vectors in $\mathbb { R } ^ { n }$ . Define the span $V = < v _ { 1 } , . . . , v _ { k } >$ of the vectors. Prove that there is a subset of the vectors $v _ { 1 } , . . . , v _ { k }$ that forms a basis for V.

6. For the cubic polynomial $x ^ { 3 } - 3 x + 2$ , use the cubic formula described below to find the root 1. [The roots of this cubic are 1 (with multiplicity 2) and -2. You do not have to prove the validity of the cubic formula.]

For a cubic polynomial $x ^ { 3 } + p x + q$ , the roots can all be found by the following formula, carried out in the complex number system. Let s be a square root of $q ^ { 2 } / 4 + p ^ { 3 } / 2 7 .$ set $A = - q / 2 + s ,$ s, and then let c be a cube root of A. Then $c - p / 3 c$ is a root of $x ^ { 3 } + p x + q$

7. Let k and n be positive integers with $k \leq n$ Give the definitions for the permutations and for the combinations of k elements from an n-element set, and state formulas for the numbers of these. Derive a formula for the number of one-to-one functions from a k-element set to an n-element set.

8. (a) State the Cauchy integral formula and use it to evaluate the complex line integral: $\oint _ { \gamma } z ^ { 2 } / ( z - i ) d z$ , where γ is a circle centered at i and oriented counterclockwise. (b) Use a parametrization of the path γ to express the line integral explicitly in terms of real integrals, with limits of integration. [You do not have to evaluate these real integrals.]