Check that B = {1, 2x + 1, (x - 1)2} is a basis for P₂, the vector space of polynomials with real coefficients of degree ≤ 2.
See Answer →Let S be any non-empty set and let V(S) be the set of all real valued functions on R. Define addition on V(s) by (f+g)(x) = f(x) + g(x) and scalar multiplication by (af)(x) = af (x). Check that (V(S), +, -) is a vector space.
See Answer →Find the signature of the quadratic form
Find adjoint of the linear operator T: C2 → C2 defined by T (z1, z2) = (z2, z1 + iz2) with respect to the standard inner product on C2.
See Answer →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.
See Answer →Check whether is an eigenvector for the matrix
What is the corresponding eigenvalue?
Verify Cayley-Hamilton theorem for the matrix
Let A be a 2 x 3 matrix, B be a 3 x 4 matrix and C be a 3 x 2 matrix and D be a 3 x 4 matrix. Is AB + CtD defined? Justify your answer.
See Answer →Find the matrix of the linear transformation T: R2 → R2 defined by T(x, y) = (2y, x - y) with respect to the ordered basis {(0, -1), (-1,0)}.
See Answer →Describe the linear transformation T:R2 → R2 such that
where B is the standard basis of R2.
See Answer →Find the kernel of the linear transformation T: R2 → R2 defined by T(x, y) = (2x + 3y, 2x – 3y).
See Answer →If {U1, U2} is an ordered basis of R2 and {f₁ (v), f2 (v)} is the corresponding dual basis find f1 (2v1+v2) and f2 (U1-2v2).
See Answer →Check whether T: R2 → R2, defined by T(x, y) = (-y, x) is a linear transformation.
See Answer →Check whether the set of vectors {1 + x, x + x², 1 + x³} is a linearly independent set of vectors in P3, the vector space of polynomials of degree ≤ 3.
See Answer →Check whether W = {(x, y, z) ∈ R³ |x + y - z = 0} is a subspace of R³.
See Answer →Find the vector equation of the plane determined by the points (1,0,-1), (0, 1, 1) and (-1,1,0).
See Answer →Find the angle between the vectors √2i + 2j + 2k and i + √2j + √2k.
See Answer →Write a detailed note on the rise and development of American Drama.
See Answer →Discuss the poetic achievements of Walt Whitman with suitable examples from the poems prescribed in the course.
See Answer →Write a critical comment on the genre of The Scarlet Letter.
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