Linear Algebra Answers

Consider the linear operator T : C 3 → C 3 , defined by T (z1,z2,z3) = (z1 −iz2,iz1 +2z2 +iz3,−iz2 +z3). i) Compute T ∗ and check whether T is self-adjoint. ii) Check whether T is unitary.?

Which of the following are binary operations on S = {x ∈ R | x > 0}? Justify your answer. i) ii) The operation dened by xy=|ln(xy)| where lnx is the natural logarithm. The operation dened by xy=x2+y3

Reduce the conic x2+6xy+y2-8=0 to standard form.

if the matrix [ 0 6-5x ] is symmetric, find the value of x.
x^2 x+3

i) The operation 5 defined by x5y = jln(xy)j where ln x is the natural
logarithm.
ii) The operation 4 defined by x4y = x2+y3.
Also, for those operations which are binary operations, check whether they are
associative and commutative

Reduce the conic x2+6xy+y2

Which of the following statements are true and which are false? Justify your answer with
a short proof or a counterexample.
i) R2 has infinitely many non-zero, proper vector subspaces.
ii) If T : V !W is a one-one linear transformation between two finite dimensional
vector spaces V andW then T is invertible.
iii) If Ak = 0 for a square matrix A, then all the eigenvalues of A are zero.
iv) Every unitary operator is invertible.
v) Every system of homogeneous linear equations has a non-zero solution

Check whether the following system of equations has a solution.
4x+2y+8z+6w = 3
2x+2y+2z+2w = 1
x+3z+2w = 3

Find inverse of the matrix B in part a) of the question by finding the adjoint as well
as using Cayley-Hamiltion theorem.

Let P3 be the inner product space of polynomials of degree at most 3 over R with
respect to the inner product
h f ;gi =
Z 1
0
f (x)g(x)dx:
Apply the Gram-Schmidt orthogonalisation process to find an orthonormal basis for
the subspace of P3 generated by the vectors (8)

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