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The probability that a ticketless traveler is caught during a trip...

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 Question:The probability that a ticketless traveler is caught during a trip is 0.1. If the traveler makes 4 trips , the probability that...

Linear Transformation Part II – Inverse Linear Transformation and Isomorphism

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Definition: A function $T$ from $V$ into $W$ is called invertible if there exists a function $S$ from $W$ to $V$ such that $TS$ is...

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Cayley Hamilton Theorem

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Statement:   Let T be a linear operator on a finite dimensional vector space V. If  f is the characteristic polynomial for T, Then $f(T)=0$, i.e....

Out of 7 consonants and 4 vowels, how many words of 3 consonants and...

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Question: Out of 7 consonants and 4 vowels, how many words of 3 consonants and 2 vowels can be formed? A. 25200               B....

Two circles $k_1$ and $k_2$ intersect in such a way that outside their intersection...

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Question:Two circles $k_1$ and $k_2$ intersect in such a way that outside their intersection is 10% of the area of the circle $k_1$ and...

Linear Functionals, Annihilator and Double Dual

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Definition:If $V$ is a vector space over the field $F$ and $S$ be a subset of $V$, the annihilator of $S$ is $S^0$...

A metric space (X, d) is connected iff every continuous function is constant

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Let (X, d) be a connected metric space and suppose that there exists a continuous function $f:X\to {\lbrace0, 1\rbrace}$. Claim: f is constant.

Three boys go to the same school. Every boy needs two cubes of the...

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Question:Three boys go to the same school. Every boy needs two cubes of the same color to play a game, but the cubes...

Every continuous function f defined on ([a, b]) is Riemann integrable

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Since, every function defined on closed and bounded interval is uniformly continuous function on [a, b], $\therefore$ By definition of uniformly continuous function,

A bounded function ​( f:[a, b]to mathbb{R} )​ is Riemann integrable iff for every...

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Suppose f is Riemann integrable on [a, b]. Let $\epsilon>0$ be arbitrary. $\int\limits_\underline{a}^bf(x)dx=\int\limits_a^\underline{b}f(x)dx$     ..... (1)

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First isomorphism theorem for vector space, homomorphism, kernel and isomorphism of vector space

Let U & V be the two real vector spaces then a mapping $f:U\to V$ is said to be homomorphism (Linear Transformation) if  (i) $f(\alpha+\beta)=f(\alpha)+f(\beta), \forall \alpha, \beta\in U$  (ii) $f(c\alpha)=cf(\alpha), \forall c\in\mathbb{R}$ & $\alpha\in U$ 

Every continuous function f defined on ([a, b]) is Riemann integrable

Since, every function defined on closed and bounded interval is uniformly continuous function on [a, b], $\therefore$ By definition of uniformly continuous function,

Dimension theorem of a quotient space: If W be a subspace of a finite dimensional vector space V over ​( mathbb{R} )​ then ​(...

Let m be the dim W.  $S=\{\alpha_1, \alpha_2, \alpha_3, ... , \alpha_m\}$ be the basis of subspace W.

Cayley Hamilton Theorem

Statement:   Let T be a linear operator on a finite dimensional vector space V. If  f is the characteristic polynomial for T, Then $f(T)=0$, i.e....

Any two closed subset of metric space are connected iff they are disjoint

Let (X, d) be a metric space and A & B are any two closed subsets of X. Let A & B are separated sets. Claim: A & B are disjoint.
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Question:The product of 2 numbers is 1575 and their quotient is $frac{9}{7}$. Then the sum of the numbers isa.   74b.   78c.   80d.   90Answer:Let...
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