Linear Algebra Answers

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)

given that a=[
1 0 3
2 1 -1
1 -1 1 ],compute the value of (a^6-5a^5+8a^4-2a^3-9a^2+31a-36 I ) using cayley-hamilton theorem

Find a matrix A whose minimal polynomial is t^3-5t^2+6t+8