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Dimension theorem of a quotient space: If W be a subspace...

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

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....

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A metric space (X, d) is disconnected iff there exists a non-empty proper open...

0
Let (X, d) be a metric space and suppose that it is disconnected. Claim: $\exists$ a non-empty proper subset of X which is both open & closed. Since, X is disconnected by definition, $\exists$ non-empty sets A & B such that $X=A\cup B$, $\bar{A}\cap

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.

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 finite dimensional vector spaces over the same field are isomorphic iff they are...

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Let $U$ and $V$ be two finite dimensional real vector spaces which are isomorphic. i.e. $\exists$ a function $f:U\to V$ which is one-one, onto and linear transformation.

Cayley Hamilton Theorem

0
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....

A woman starts shopping with Rs. X and Y paise, spends Rs. 3.50 and...

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Question:A woman starts shopping with Rs. X and Y paise, spends Rs. 3.50 and is left with Rs. 2Y and 2X paise. The...

Linear Transformation Part I – Algebra of Linear Transformation

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Theorem:  Let V and W be two vector spaces on the same field. Let S and T be linear transformations from V into W. Then...

Every monotonic function on [a, b] is Riemann integrable

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Theorem:  Every monotonic function on is Riemann integrable. Proof: Let ​( f:to mathbb{R} )​ be monotonic function. If f is constant function then obviously it is Riemann...

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A metric space (X, d) is disconnected iff there exists a non-empty proper open & closed subset of X which is both open and...

Let (X, d) be a metric space and suppose that it is disconnected. Claim: $\exists$ a non-empty proper subset of X which is both open & closed. Since, X is disconnected by definition, $\exists$ non-empty sets A & B such that $X=A\cup B$, $\bar{A}\cap

First and Second fundamental theorem of calculus

If $F'(x)=f(x)$ then F is called primitive or antiderivative of f. e.g. $F(x)=x^2sin(\frac{1}{x})$ $\therefore F'(x)=2xsin(\frac{1}{x})+x^2.cos(\frac{1}{x}).(\frac{-1}{x^2})$ 

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.

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.
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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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