Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) If W₁ and W₂ are proper subspaces of a non-zero, finite dimensional, vector space V and dim(W₁) > dim(V)/2, dim(W2) > dim(V)/2, the W₁ ∩ W₂ ≠ {0}.
ii) If Vis a vector space and S = {U1, U2,..., Un} CV, n ≥ 3, is such that v₁ ≠ v; if i ≠ j, then S is a linearly independent set.
iii) If T1, T2: V → Vare linear operators on a finite dimensional vector space Vand T1 ∘ T2 is invertible, T2 ∘ T₁ is also invertible.
iv) If an n x n square matrix, n ≥ 2 is diagonalisable then it has the same minimal polynomial and characteristic polynomial.
If T1, T2: V → Vare self adjoint operators on a finite dimensional inner product space V, then T₁ + T₂ is also a self adjoint operator.
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Let V be the vector space of all real valued functions that are twice differentiable in R and
S = {cos x, sin. x, x cos x, x sin x}.
Check that S is a linearly independent set over R. (Hint: Consider the equation
a 0cos x + a₁ sin x + a2x cos x + a3x sin x.
(Put x = 0, π, π/2,π/4, etc. and find a₁.)
मान लीजिये कि A एक 2 x 3 आव्युह है, B एक 3 x 4 आव्यूह है, C एक 3 x 2 आव्यूह है और D एक 3 x 4 आव्यूह है। क्या AB + CtD परिभाषित है? अपने उत्तर की पुष्टि कीजिये।
मान लीजिये कि S कोई भी एक अरिक्त समुच्चय है और V (S), सपर सभी वास्तविक मान वाले फलनों का समुच्चय है। V(S) पर योग (f+g)(x) = f(x) + g(x) द्वारा परिभषित कीजिये और आदिश गुणन ( f) (x) =
f(x) द्वारा परिभषित कीजिये। जाँच कीजिये कि V (S), +, .) एक सदिश समष्टि है।
यदि का एक क्रमित आधार है और
इस का संगत द्वैत आधार है तो
और
निकालिये।
द्विघाति समघात −?2 + ?2 + ?2 + 4?? + 4?z का लाम्बिक विहित समानयन और इस के मुक्य अक्ष निकालिये।
Check whether is an eigenvector for the matrix
What is the corresponding eigenvalue?
Let C[0, 1] be the inner product space of continous real valued functions on the interval [0, 1] with the inner product
Find the inner product of the functions f(t) = 2t,g(t) = 1/ t2+5.