Check that given by
is well defined and 1 - 1.
Further, check that
for any
.
The function ( g:\mathbb{R}\setminus{2} \to \mathbb{R} ) is defined by ( g(x) = \frac{x-1}{2-x} ). First, to check if ( g ) is well-defined: the domain is ( \mathbb{R} \setminus {2} ), so ( x \neq 2 ). The denominator ( 2-x ) is nonzero for all ( x \neq 2 ). Therefore, ____ _______ _______ _______ _______ __________ ________ _____ _________ ________ ___.
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Find the solutions to the following system of equations by reducing the corresponding augmented matrix to row-reduced echelon form.
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
.
जाँच कीजिए कि के लिए
और
के लिए
।
Find the values of for which the matrix
is Hermitian.
Check that f(x) is well defined and 1 - 1.
Is the matrix of the linear operator T non-singular? Justify your answer.
) 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.