Linear Algebra Answers

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

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.

. Define the kernel and the image of a linear transformation M : V → W and hence Show that the Kernel and image of M are vector subspaces of V and W respectively. 

Define a vector subspace of a vector space V

Show that if P and Q are vector Subspaces of a vector space V then P ∩ Q is also a vector subspace of V.

Define a vector subspace of a vector space V

Define the kernel and the image of a linear transformation K : M → N and hence Show that the Kernel and image of K are vector subspaces of M and N respectively

Find a 2×2 matrix A such that A^2 is a diagonal but not A

18. Suppose R^2 has weighted inner product given as <u, v> =(3u base 1 v base 1 + 2u base 2 V base 2 for u = (u base 1, u base 2), v = (V base 1, v base 2). Let u = (1, 2), v = ( 2, - 1) and K = 3. Then the valued of <u, kv> is.....

(i) 4

(ii) 6

(iii) 18

(iv) None

19. Suppose that u, v "\\isin" V are such that ||u|| = 2, ||u +v|| = 3 and ||u - v|| = 4. Then ||v|| is?

(i) 17/2

(ii) √17

(iii) Does not exist

(iv) None

20. For a given matrix A="\\begin{bmatrix}\n3 & 1 \\\\\n 1 & 3\n\\end{bmatrix}", the matrix P that is orthogonally diagonalizes A is of the following matrices are diagonalisable

(i)P= "\\begin{bmatrix}\n1\/\u221a2 & 1\/\u221a2 \\\\\n 1\/\u221a2 & - 1\/\u221a2\n\\end{bmatrix}"

(ii)P= "\\begin{bmatrix}\n 0 & 1 \\\\\n 1 & 0 \n\\end{bmatrix}"

(iii)P="\\begin{bmatrix}\n-1\/\u221a2 & 1\/\u221a2 \\\\\n - 1\/\u221a2 & 1\/\u221a2\n\\end{bmatrix}"