Calculus : IITJEE Advanced Math tutorial problems: Part 1

Problem 1: Prove that $|x| \leq \sum_{i=1}^{n} |x^{i}|$

Problem 2: When does equality hold in the following theorem? $|x+y| \leq |x|+|y|$ ? Hint: Re-examine the proof of the theorem, the answer is not “when x and y are linearly dependent.”

Problem 3: Prove that $|x-y| \leq |x|+|y|$ . When does inequality hold?

Problem 4: Prove that $||x|-|y|| \leq |x-y|$ ?

Problem 5: The quantity $|y-z|$ is called the distance between x and y. Prove and interpret geometrically the “triangle inequality” : $|x-z| \leq |x-y|+|y-z|$ .

Problem 6: Let functions f and g be integrable on $[a,b]$ .

(a) Prove that $|\int_{a}^{b}| \leq (\int_{a}^{b}f^{2})^{\frac{1}{2}}.(\int_{a}^{b}g^{2})^{\frac{1}{2}}$ . Hint: Consider separately the cases $0 = \int_{a}^{b}(f-g \lambda)^{2}$ for some $\lambda \in \Re$ and $0 < \int_{a}^{b}(f-g\lambda)^{2}$ for all $\lambda \in \Re$ .

(b) If equality holds, must $f=g \lambda$ for some $\lambda \in \Re$ ? What if f and g are continuous?

(c) Show that the following theorem is a special case of (a) above: $|\sum_{i=1}^{n}x^{i}y^{i}| \leq |x|.|y|$ , equality holds if and only if x and y are linearly dependent.

Problem 7: A linear transformation $T: \Re^{n} \rightarrow \Re^{n}$ is norm preserving if $|T(x)|=|x|$ amd inner product preserving if $<Tx, Ty> = <x,y>$

(a) Prove that T is norm preserving if and only if T is inner product preserving.

(b) Prove that such a linear transformation T is $1-1$ and $T^{-1}$ is of the same sort.

Problem 8:

If $x, y \in \Re^{+}$ are non-zero, the angle between x and y, denoted $\angle {(x,y)}$ is defined as $\arccos{(\frac{<x,y>}{|x|.|y|})}$ , which makes sense by the following theorem :

$<x,y> \equiv |\sum_{i=1}^{n}x^{i}y^{i}| \leq |x|.|y|$

The linear transformation T is angle preserving if T is 1-1, and for $x,y \neq 0$ we have $\angle {(Tx,Ty)} = \angle{(x,y)}$

(a) Prove that if T is norm preserving, then T is angle preserving.

(b) If there is a basis $\{ x_{1}, x_{2}, \ldots, x_{n}\}$ of $\Re^{n}$ and numbers $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$ such that $Tx_{i}=\lambda_{i} x_{i}$ , prove that T is angle preserving if and only if all $\lambda_{i}$ are equal.

Problem 9: If $0 \leq \theta < \pi$ , let $T: \Re^{2} \rightarrow \Re^{n}$ have the matrix $\left | \begin{array}{cc} \cos{\theta} & \sin{\theta} \\ -\sin{\theta} & \cos{\theta} \end{array} \right |$ .

Show that T is angle preserving and if $x \neq 0$ , then $\angle{(x, Tx)}= \theta$

Problem 10: If $T: \Re^{m} \rightarrow \Re^{n}$ is a linear transformation, show that there is a number M such that $|T(h)| \leq M|h|$ for $h \in \Re^{m}$ . Hint: Estimate $|T(h)|$ in terms of $|h|$ and the entries in the matrix of T.

Problem 11: If $x, y \in \Re^{n}$ and $z, w \in \Re^{m}$ , show that $<(x,z),(y,w)> = <x,y>+<z,w>$ and $|(x,z)|= \sqrt{|x|^{2}+|z|^{2}}$ . Note that $(x,z)$ and $(y,w)$ denote points in $\Re^{n+m}$ .

Problem 12: Let $(\Re^{n})^{*}$ denote the dual space of the vector space $\Re^{n}$ . If $x \in \Re^{n}$ , define $\phi_{x} \in (\Re^{n})^{*}$ by $\phi_{x}(y)=<x,y>$ . Define $T: \Re^{n} \rightarrow (\Re^{n})^{*}$ by $T(x)=\phi_{x}$ . Show that T is a 1-1 linear transformation and conclude that every $\phi \in (\Re^{n})^{*}$ is $\phi_{x}$ for a unique $x \in \Re^{n}$ .

Problem 13: If $x, y \in \Re^{n}$ , then x and y are called perpendicular (or orthogonal) if $<x,y>=0$ . If x and y are perpendicular, prove that $|x+y|^{2} = |x|^{2}+|y|^{2}$ .

Regards,

Nalin Pithwa

For the love of Maths: the Queen of all Sciences

Calculus : IITJEE Advanced Math tutorial problems: Part 1

Leave a Reply Cancel reply

Share this:

Like this:

Related

Leave a Reply Cancel reply