द्वारा परिभाषित फलन
लीजिए:
प्रश्न अधूरा है। केवल फलन ( f(x)=\dfrac{2x+1}{x+1} ) परिभाषित किया गया है, _______ ____ _________ ________ ___ __________.
__________ ________ _________ _________ _________.
_________ ____ __________ ______ _______ _____ ________ ________ ___ __________.
________ __________ _________ _________ ________ ___ _______.
_____ _____ ____ ________ __________ ________ _____.
____ _____ _____ _____ _____ ________ ____ ______ ______ _______ _______.
_____ ______ _____ _______ ______ ____ __________.
_________ ________ ___ _____ _________ ______ __________ ____ ___ _______.
_______.
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जाँच कीजिए कि पर रैखिकता: स्वतंत्र समुच्चय है। (संकेत: समीकरण
लीजिए।
इत्यादि रखिए और ai के हल कीजिए)
Show that, if A is any matrix with real entries, then there is a
symmetric matrix S and a
skew symmetric matrix S' such that
.
) Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) The function defined by
is 1-1.
ii) The operation $ defined byx y = \log(xy)is a binary operation onS, whereSis the set\{x \in \mathbf{R} \mid x > 0\}$.
iii) The set is a subspace of
.
iv) There is no matrix of rank 6.
v) If V and V' are vector spaces and is a linear transformation, then whenever
are linearly independent,
are also linearly independent.
vi) If V is a vector space and is a linear operator with
, then T is not diagonalisable.
vii) The degree of the minimal polynomial of a matrix is at most 2.
viii) For any matrix A,
.
ix) The only matrix which is both symmetric and skew-symmetric is the zero matrix.
x) There is no co-ordinate transformation that transforms the quadratic form x2 + y2 + z2 to the quadratic form xz + yz.
जाँच कीजिए कि पर रैखिकता: स्वतंत्र समुच्चय है। (संकेत: समीकरण
Check that f(x) is well defined and 1 - 1.
द्वारा परिभाषित फलन
लीजिए:
जाँच कीजिए कि के लिए
और
के लिए
।