Linear Algebra Answers

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

If T:R^3 tends to R^3 is defined by T(x, y, z)=(2x+y-2z, 2x+3y-4z, x+y-z) Find all eigen value of T and find a basis of each eigen space. Is T is diagonalizable?

Let T: R^5 tends to R^3 defined by T(a, b, c, d, e)=(a+2b+2c+d+e, a+2b+3c+2d-e, 3a+6b+8c+5d-e). Find the basis and the dimension of the kernel and the image of T

Let W = {(x, y) E R² | 2x + 3y = 0} Show
that W is a subspace of R2 . Find the
dimension of W. Also show that the cosets of
W are lines 2x + 3y + c = 0, where c belongs to R

Using Cayley — Hamilton theorem, evaluate A^(7) if A=[ 3 2
-1. 0 ]

Let V be the real vector space of polynomials
over R of degree at most 2 and W be the
subspace of V generated by (1+x², 1+2x² }.
Find the kernel of the differential linear
operator d/dx on W

Let W {(x, y, z) R³ : x + y + z = 0}. Check
if W is a subspace of R³. Find a non-zero
subspace U of R3 so that W(intersection)U = (0)

Find a 2×2 singular matrix A that maps
(1, 1)^T into (1, 3)^T

Give an example of a non linear transformation T:R^2 into R^2 such that
T^-1 (0)=0 but T is not one - one.