# PreLiminary Exam, Spring 2004(3 HOURS, 8 PROBLEMS COUNTED EQUALLY)

1. Suppose $f , g : A \longrightarrow A$ are functions with f o g injective.

aProve that g must be injective.

b) Give an example to show that f need not be injective.

2. a) Give the definition for a function $L : \mathbb { R } ^ { n }  \mathbb { R } ^ { m }$ to be a linear transformation.

Prove that the kernel of a linear transformation $L : \mathbb { R } ^ { n }  \mathbb { R } ^ { m }$ is a subspace of $\mathbb { R } ^ { n }$

3. a) Define what it means for vectors $\boldsymbol { v } _ { 1 } , . . . , \boldsymbol { v } _ { n }$ in a vector space V to be linearly independent.

b) Suppose that vectors $v _ { 1 } , . . . , v _ { n }$ in a vector space V span V and that no proper subset of $\{ v _ { 1 } , . . . , v _ { n } \}$ spans V. Prove that $v _ { 1 } , . . . , v _ { n }$ are linearly independent.

4. Diagonalize the matrix $A = \left[ { \begin{array} { r r } { 1 } & { 2 } \\ { 4 } & { - 1 } \end{array} } \right]$ and use the diagonal form of A to compute $A ^ { 2 }$

5. Suppose L is a real number and $f , g : \mathbb { R } \to \mathbb { R }$ Prove that if $\begin{array} { r } { \operatorname { i m } _ { x  0 } g ( x ) = L } \end{array}$ and f is continuous at $\scriptstyle { L , }$ then $\scriptstyle \operatorname* { l i m } _ { x \to 0 } f ( g ( x ) )$ also exists.

6. Define a sequence $\left( x _ { n } \right)$ by $x _ { 1 } = 1$ and $\begin{array} { r } { x _ { n + 1 } = \frac { 1 } { 4 } x _ { n } ^ { 2 } + 1 } \end{array}$ for $\pmb { n } \in \mathbb { N } .$

a)Give an inductive proof that $x _ { n } \leq 2$ for all $\boldsymbol { n } \in \mathbb { N }$

Prove that $x _ { n + 1 } \geq x _ { n }$ for each $\displaystyle n \in \mathbb { N } .$ Hint: Consider $x _ { n + 1 } - x _ { n } . \rfloor$

7. Suppose f is a continuous function satisfying the equation $f ( x ) = 5 + \int _ { 0 } ^ { x } 3 f ( t ) d t$ for all real x. Argue that f must be differentiable and then find al such function(s) explicitly.

8. Provide examples of the following.
   [No justification is required.]

a) a true implication whose converse is false,

a function $f : \mathbb { R } \to \mathbb { R }$ which is continuous at 7, but not differentiable there,

a function $f : \mathbb { R } ^ { 2 } \to \mathbb { R }$ which is continuous in each variable separately, but is not continuous at (0, 0),

da bounded sequence which is not Cauchy,

e)a (real) power series whose domain of convergence is the closed interval [0, 2].