Linear Algebra Answers

Which of the following are binary operations on S = {x ∈ R | x > 0}? Justify your answer. i) The operation defined by xy=|ln(xy)| where lnx is the natural logarithm. ii.)The operation ∆ defined by x∆y=x2+y3. Also, for those operations which are binary operations, check whether they are associative or commutative.

Let T:P2→P1 be defined by T(a+bx+cx2)=b+2c+(a−b)x. Check that T is a linear transformation. Find the matrix of the transformation with respect to the ordered bases B1={x2,x2+x,x2+x+1} and B2={1,x}. Find the kernel of T.

Verify that W is a subspce of V. Assume that V has the standard operations.

W={(x1,x2,x3,0)} where x1,x2,x3 are real numbers.
V=R4

Let P^(e)(x) = {p(x) ∈ R[x]|p(x) = p(−x)} Find W =P(e)∩P3. Find a basis for W and the dimension of W.

Find the inverse of the matrix  −1 2 1 0 1 1 1 0 2  using row reduction.

Define T:R3→R3 by T(x,y,x)=(x+y,y,2x−2y+2z).Check that T satisfies the polynomial(x−1)2(x−2).Find the minimal polynomial of T

Find the orthogonal canonical reduction of the quadratic form x2+y2+z2−2xy−2xz−2yz.Also,find its principal axes.

Let T : R³→ R³ be defined by T (x1, x2, x3) = (x1 −x3, x2 −x3, x1). Is T invertible? If yes, find a rule for T–¹ like the one which defines T ?

Let T : P2-P1 be defined by T(a+bx+cx2) = b+2c+(a-b)x: Check that T is a linear transformation. Find the matrix of the transformation with respect to the ordered bases B1 = {x2,x2+x,x2+x+1} and B2 = {1,x}. Find the kernel of T.

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.?