Linear Algebra Answers

Solve by finding the basis over R for the solution space.

(A) X + 3y -3z=0

2x - 3y + z=0

3x -2y + 2z=0

(B) X + Y + Z + W=0

2X + 3Y - Z +W=0

3X + 4Y +2W=0

A certain polynomial has a graph which has an end behaviour of II → IV. It has 2 turning points and 1 x-intercept. Sketch the shape of the graph and indicate what degree the polynomial is and whether its leading coefficient is positive or negative.

Find the vector equation of the plane determined by the points (1,-2,1),(1,0,1) and (1,-1,1). Also check whether (1/2,1/2,1/2) lies on it.

Find the minimal polynomial of T: R^3→R^3 defined by T(a,b,c)= (a-b,b,c)

Let V= R^3

W={(x1, x2, x3)| x1-x2 =x3}. Show that W is a

subspace of V. Further,find a basis for W and hence,find the dimension of W.

1.What is affine space, affine set and affine varieties.Explain with examples.

2.Define a relation between affine space and vector space with examples

To determine the eigenvalues and eigenvectors of the following matrix :

2 -2 0

P = -2 1 -2

0 -2 0

To verify Cayley-Hamilton theorem for

1 2 3

M = 2 -1 4

3 1 1

37. Reduce the following Quadratic form to Canonical form by Orthogonal transformation

(ii) Q= 2x 1x2+2x2x3+2x3x1

Let 0<θ<2π, θ ≠ π. Consider the linear transformation T: C^2→C^2 given by matrix

[ cosθ -sinθ](w.r.t standard basis)

[ sinθ cosθ]. Find the vector v1, v2 such that Tv1= e^iθv1, Tv2= e^-iθv2. Is {v1,v2} a basis for C^2? Give reason for your answer.