1. Who discovered Zero (0)?
Answer:Â Aryabhatta, AD 458
Explanation: Aryabhatta invented zero but he didn’t give any symbol for zero, Brahmagupta was the first to give symbol for zero and rules to compute with zero.
If ​\( F'(x)=f(x) \)​ then F is called primitive or antiderivative of f.
e.g.\( F(x)=x^2sin(\frac{1}{x}) \)
​\( \therefore F'(x)=2xsin(\frac{1}{x})+x^2.cos(\frac{1}{x}).(\frac{-1}{x^2}) \)​
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.
1. Who discovered Zero (0)?
Answer:Â Aryabhatta, AD 458
Explanation: Aryabhatta invented zero but he didn’t give any symbol for zero, Brahmagupta was the first to give symbol for zero and rules to compute with zero.
If V is a vector space over the field F and S be a subset of V, the annihilator of S is ​\( S^0 \)​ and ​\( S^0 \)​ is the set of linear functionals f on V such that ​\( f(\alpha)=0 \)​ for each ​\( \alpha \in S \)
A function T from V into W is called invertible if there exists a function S from W to V such that TS is an identity on W and ST is an identity on V.Â
i.e. ​\( TS=I_w, ST=I_v \)
Let ​\( \alpha, \beta \in V \)​ and c \in F and S & T are linear transformation from V to W.Â
​\( (S+T)(c\alpha+\beta) \)
\( =S(c\alpha+\beta)+T(c\alpha+\beta) \)
Let K be the commutative ring with identity consisting of all polynomials in T.
Choose an ordered basis ​\( \{\alpha_1, \alpha_2, ... ,\alpha_n\} \)​ for V.
Let A be the matrix of T in the basis ​\( \{\alpha_1, \alpha_2, ... ,\alpha_n\} \)
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