Graduate Preliminary Examination, Spring 2003

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

Instructions: Work 7 of the 8 problems.

#1. (a) Define the span of vectors $v _ { 1 } , . . . , v _ { k }$ in $\mathbb { R } ^ { n }$

$$
v _ { 1 } , . . . , v _ { k } ,
$$

$$
v _ { 1 } , . . . , v _ { k }
$$

$$
v _ { 1 } , . . . , v _ { k } , w
$$

#2. Does there exist a $\Im \times 2$ complex matrix A such that

$$
A ^ { t } A = I _ { 2 } , { \mathrm { ~ a n d ~ } } A A ^ { t } = I _ { 3 } ?
$$

If ${ \mathfrak { s o } } ,$ giv an example; if not, prove it.
Here Ik is the 60 $I _ { k }$ $k \times k$ identity matrix.

#3. Prove the formula $\textstyle \sum _ { i = 1 } ^ { n } i ^ { 2 } = { \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } }$

#4. Prove that $\pmb { x } >$ ln x for all ${ \pmb x } > { \pmb 0 } .$

#5. Prove that any linear function $L : \mathbb { R } ^ { 2 } \to \mathbb { R } ^ { 2 }$ is continuous.

#6. Does there exist a divergent series $\textstyle \sum _ { i = 0 } ^ { \infty } a _ { i }$ of positive real numbers $\pmb { a _ { i } }$ such that $\sum _ { i = 0 } ^ { \infty } \sqrt { a _ { i } }$ converges?
If so, give an example; if not, prove it.

#7. (a) Let f be a real-valued function defined on Rn.
Define differentiability of f at a $\mathbb { R } ^ { n }$ $\pmb { f }$ point p.

(b) Show that the function $f : \mathbb { R } ^ { 2 } \to \mathbb { R }$ defined by

$$
f ( x , y ) = { \frac { x y } { x ^ { 2 } + y ^ { 2 } } } { \mathrm { ~ f o r ~ } } ( x , y ) \neq ( 0 , 0 ) , { \mathrm { ~ a n d ~ } } f ( 0 , 0 ) = 0
$$

is not differentiable at (0, 0).

#8. Consider the line integral $\textstyle \int _ { C } 2 x d x + x ^ { 2 } y d y ,$ , where C is the boundary of the unit square in the 1st quadrant.
Evaluate the line integral (a) directly and (b) by using Green's theorem.