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.

(c) What are all angle preserving T: \Re^{n} \rightarrow \Re^{n}?

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}.


Nalin Pithwa

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