Linear Algebra Answers

Find an orthonormal basis of R^3 of which (1√10,0,-3/√10) is one element

R^3 is a inner product space over the inner product

<(x1,x2,X3),(y1,y2,y3)> = x1y1+ x2y2 - x3y3

True or false with full explanation

If T: U to V is a one- one linear transformation between finite- dimensional vector space V and W , then T is invertible. True or false with full explanation

Check that T = R^3 to R^3, defined by

T(x1,x2,X3)= (x1+X3, x2+2x3, x1-x2-x3) is a linear operator. Also find the kernel

For any two subspace W1,W2 of R^3 of dimension 2, W1+ W2 is a direct sum . True or false with full explanation

If some eigenvalues of a matrix are repeated, the matrix is not diagonisable.true or false with full explanation

R^3 has infinitely many non zero, proper vector subspaces. True or false with full explanation

Let T: R^3 to R^3 be the linear transformation defined by

T(x,y,z)= (-x,x-y, 3x+2y+z)

Check whether T satifies the polynomial (x-1)(x+1)^2. Also the find of minimal polynomial of T.

Check that {1,(x+1),(x+1)^2} is a basis of the vector space of polynomial over R of degree at most 2. Find the coordinate of 3+x+2x^2 with respect to the basis.

Let P = [ -1 4 5] . Determine P^-1 using

[ 0 2 -3]

[ 0 0 8]

Cayley- Hamilton theorem. Further use P^-1 to express (x1, x2, x3) in terms of (-1,0,0), (4,2,0), ( 5,-3,8)