Mathematics
The probability that a ticketless traveler is caught during a trip...
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
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
Statement:Â Â
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...
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...
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
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
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...
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
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...
Suppose f is Riemann integrable on [a, b].
Let $\epsilon>0$ be arbitrary.
$\int\limits_\underline{a}^bf(x)dx=\int\limits_a^\underline{b}f(x)dx$Â Â Â ..... (1)
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