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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.
Define: R^3→R^3 by
T(x,y,z)=(x-y+z,x+y,y+z)
Let v1= (1,1,1), v2= (0,1,1), v3= (0,0,1). Find a matrix of T with respect to the basis {v1,v2,v3}. Futher check T is invertible or not.
Complete {(2,0,3)} to form an orthogonal basis of R^3
