Check that given by
is well defined and 1 - 1.
Further, check that
for any
.
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) 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.
Let and let
be the function defined by
.
Write down the matrix of T on W w.r.t the basis S.
जाँच कीजिए कि पर रैखिकता: स्वतंत्र समुच्चय है। (संकेत: समीकरण
लीजिए।
इत्यादि रखिए और ai के हल कीजिए)
Find the inverse of the matrix B in part a) by using Cayley-Hamilton theorem.
जाँच कीजिए कि किसी के लिए
है।
जाँच कीजिए कि द्वारा दिया गया
सुपरिभाषित और 1-1 है। इसके आगे जाँच कीजिए कि किसी
के लिए
है।