Linear Algebra Answers

Determine the polynomial function whose graph passes through the points (2, 4), (3,6) and (5,10). Also sketch the graph of the polynomial function. (Using Cramer’s Method).

Let T: R^2 to R^3 be defined by (a, b)=(a+b, a-2b, 3a+b). Show that T is nonsingular. Hence find T inverse.

Find a linear mapping T: R^3 tends to R^3, whose image is spanned by (1, 2, 3),
(4, 5, 6)

Show that
Rank(ST)=Rank s ,if T is non singular
Where S,T : V->V are linear transformation of a finite dimensional vector space.

Find all the roots in C of the equation x^4-2x^3+4x^2+6x-21=0, give n that two of its roots are equal in magnitude and opposite in sign.

Let T be a linear operator whose matrix with
respect to the basis {(1,1,1),(1,-1,0),(1,0,-1)} is the matrix [
4 0 0
0 1 0
0 0 1. ]
Obtain the matrix of T wrt the standard basis.

Which of the following statements are True and
which are False ? Justify your answer with a
short proof or by a counter-example.

(a) The operation *, defined by x * y = log (xy) is
a binary operation on S, where
S={xER x>0}.

(b) If a and b are eigenvalues of two n x n
matrices A and B respectively, then a + b is
an eigenvalue of A + B.

(c) If S and T are linear transformations such
that SoT is defined and is 1 — 1, then S is
1 — 1.

(d) T : R³- R³: T((x1, x2, x3), (y 1 , y2, y3)) =
(x1 + x2 + x3) . (y1 + y2 + y3)
is an inner product on R³.

(e) {India, — 5, Jamila} is a set.

If M is a singular Matrix, is there a value of k∈N for which kM will be non singular? Give reasons for your answer