क्या रैखिक संकारक T का आव्यूह व्युत्क्रमणीय है। अपने उत्तर की पुष्टि कीजिए।
रैखिक संकारक TT का आव्यूह व्युत्क्रमणीय (invertible) होना इस पर निर्भर करता है कि उसका निर्धारणांक (determinant) शून्य नहीं है। यदि एक रैखिक संकारक का निर्धारणांक शून्य होता है, तो _______ _______ _________ _______ ______ _________ _____ ______ ____ ____ ________ ___.
__________ ________ __________ ______ ______.
____ ____ _____ __________ __________ __________ __________ _____ __________.
_____ ________ ________ __________ ________ ____ _____ __________ __________.
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___ ____ ____ ___ _______.
_________ ___ _________ _____ _________ _________ ________ ____.
____ ________ _____ _______ _______ _________ ___ _______ ________ ________ __________ ____.
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__________ __________ ___ _____ ______ ________ ________ ________ _______ ____.
_____ _______ ______ __________ ____ ____ ________ ___ __________ ______.
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Write down the matrix of T on W w.r.t the basis S.
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 f(x) = cosx is 1-1.
ii) The operation * defined by x*y = log(xy) is a binary operation on S, where S is the set {x ∈ R x > 0}.
(iii)The set {(x1.x2,...,xn) X1,X2,...,xn ∈ R,X1 = 2x2+3} is a subspace of Rn
iv) There is no 7 x 5 matrix of rank 6.
v) If V and V' are vector spaces and T: V→ V' is a linear transformation, then whenever u1,u2,..., uk are linearly independent, Tu₁, Tu2, ... Tuk are also linearly independent.
vi) If V is a vector space and T: V → V is a linear operator with det(T) = 0, then T is not diagonalisable.
vii) The degree of the minimal polynomial of a 3 x 3 matrix is at most 2.
viii) For any 2 x 2 matrix A, Adj (A') = (Adj(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 V be the set of all functions that are twice differentiable in R and
S = {cosx, sinx, xcosx,xsinx}.
कैली-हैमिल्टन प्रमेय द्वारा भाग क) में आव्यूह B का व्युत्क्रम ज्ञात कीजिए।
जाँच कीजिए कि x∈ R\{2} के लिए (fog) (x) = x और x∈ R \{-1} के लिए (fog) (x) = x
Are there values of a ∈ C for which the matrix
is unitary? Justify your answer.
Check that f(x) ≠ 2 for any x ∈ R.