Linear Algebra Answers

3. Reduce the quadratic form x 1 ^ 2 +5 x 2 ^ 2 + x 3 ^ 2 +2x 1 x 2 +2x 2 x 3 +6x 3 x 1 to canonical form through an orthogonal transformation and also find its rank, index, signature and nature of the quadratic form.

2. ſi 0 Verify Cayley-Hamilton theorem, Find A ^ 4 when A= 2 1 1 1 1 -1 3 -1 1

If y is an eigenvalue of an orthogonal matrix ,show that 1/y is also it's eigenvalue

True or false and give reason for your answer


i) If all the Eigen values of a matrix are real, then the
matrix must be symmetric.
ii) If V and W are real vector spaces, with T : V --> W
being a linear transformation then V/kerT isomorphic to W
iii) There are at least three different unitary matrices of order 2.
iv) There are two subspaces U and W of R3 such that U(intersection) W is empty.
v) If the determinant of a matrix is zero, then the matrix cannot be diagonalised.

Consider the zero mapping O:U tends to V
defined by 0(v)=0 for all v belongs to V. Find the kernel and image of O

a) Let T = R2–>R3 be a linear
transformation given by :
T(x1,x2) =(x1 + X2,x1, x1-x2 )
Show that {(1,1), (1, 0)} and {(1, 1, 0), (1, 0, 1), (0, 1, 1)) are bases of R2 and R3
respectively.Find the matrix of T with respect to these ordered basis.

Let W be a solutions space of homogeneous equation 2a+3b+4c=0
Describe the cosets of W in R^3

Find the characteristic polynomial of the linear operator T:R^2 tends to R^2 defined by T(a, b)= (3a+5b, 2a-7b)