IAS (Admin.) Mains Mathematics: Questions 84 - 89 of 668
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Question 84
Appeared in Year: 2012
Describe in Detail Subjective▾
Consider the linear mapping by
Find the matrix relative to the basis and the matrix relative to the basis { (1,2) , (2,3) } . (CS main Paper 1)
EditExplanation
Given by
And
… (85 more equations) …
Question 85
Appeared in Year: 2012
Describe in Detail Subjective▾
If is a characteristic root of a non-singular matrix , then prove that is a characteristic root of . (CS main Paper 1)
EditExplanation
Since is an eigen value of a non-singular matrix
and there exist a non-zero column vector such that
… (15 more equations) …
Question 86
Appeared in Year: 2009
Describe in Detail Subjective▾
Prove that the set of the vectors in which satisfy the equations and is a subspace of . What is the dimension of this subspace find one of its bases. (CS main Paper 1)
EditExplanation
Let be the given vector space…
… (590 more equations) …
Question 87
Appeared in Year: 2015
Describe in Detail Subjective▾
Show that the function is harmonic. Find its conjugate harmonic function . Also, find the corresponding analytic function in the terms of Z. (CS main Paper 2)
EditExplanation
Given
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Question 88
Appeared in Year: 2012
Describe in Detail Subjective▾
Let be the vector space of all matrices over the field of real numbers.
Let be the set consisting of all matrices with zero determinant. Is a subspace of Justify your Answer. (CS main Paper 1)
EditExplanation
lemma: Let be a vector space over a field Then is a subspace of its and
Proof:- Let
… (731 more equations) …
Question 89
Appeared in Year: 2013
Describe in Detail Subjective▾
Let be the linear transformation defined by
Find a basis and the dimension of the image of and the Kernel of . (CS Paper I)
EditExplanation
We know that Basis of is
… (1079 more equations) …