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Linear Algebra
a)Show that, if A is any n×n matrix with real entries, then there is a n×n symmetric matrix S and a n×n skew symmetric matrix S' such that A=S+S'.(3)b)Find the solutions to the following system of equations by reducing the corresponding augmented matrix to row-reduced echelon form.
2a+3b+4c+d=8
a+2b+2c+2d=3
a−b+c+3d=3
2a+3b+4c+d=8
a+2b+2c+2d=3
a−b+c+3d=3
Linear Algebra
Apply the Gram-Schmidt diagonalisation process to find a northonormal basis for the subspace of C4 generated by the vectors {(1,i,0,1),(1,0,i,0),(−i,0,1,−1)}(6)b) Find the orthogonalcanonical reduction of the quadratic form x2−2y2+z2+2xy+6yz and its principal axes. Also, find the rank and signature of the quadratic form.
Linear Algebra
a)Find the values of a,b∈C for which the matrix (1. i. 1+i)
( a. 0. b. )
(1−i. 2+i. 1) is Hermitian.(2)b)
Are there values of a∈C for which the matrix (1. 0. 0 )
(0. −1/√2. 1/√2)
(0. 1/√2. a. )
is unitary? Justify your answer.(3)c)Let (x1,x2,x3) and (y1,y2,y3) represent the coordinates with respect to the bases B1={(1,0,0),(0,1,0),(0,0,1)},B2={(1,0,0),(0,1,2),(0,2,1)}.If Q(X)=x21+2x1x2+2x2x3+x22+x23,find the representation of Q in terms of (y1,y2,y3).
( a. 0. b. )
(1−i. 2+i. 1) is Hermitian.(2)b)
Are there values of a∈C for which the matrix (1. 0. 0 )
(0. −1/√2. 1/√2)
(0. 1/√2. a. )
is unitary? Justify your answer.(3)c)Let (x1,x2,x3) and (y1,y2,y3) represent the coordinates with respect to the bases B1={(1,0,0),(0,1,0),(0,0,1)},B2={(1,0,0),(0,1,2),(0,2,1)}.If Q(X)=x21+2x1x2+2x2x3+x22+x23,find the representation of Q in terms of (y1,y2,y3).
Linear Algebra
For the following matrices, check whether the reexists an invertible matrix P such that P−1 A Pis diagonal. When such a P exists,find P.
i)A=( 0 1 -3)
( 2 -1 6)
( 1 -1 4)
(ii) B is equal to. ( 0 0 -2)
( 1 2 1)
( 1 0 3)
b)Find the inverse of the matrix B in part (a) by using Cayley-Hamilton theorem.(3)c)Using the fact that det(AB)=det(A)det(B)for any two matrices A and B,prove the identity (a2+b2) (c2+d2)=(ac−bd)2+(ad+bc)2
i)A=( 0 1 -3)
( 2 -1 6)
( 1 -1 4)
(ii) B is equal to. ( 0 0 -2)
( 1 2 1)
( 1 0 3)
b)Find the inverse of the matrix B in part (a) by using Cayley-Hamilton theorem.(3)c)Using the fact that det(AB)=det(A)det(B)for any two matrices A and B,prove the identity (a2+b2) (c2+d2)=(ac−bd)2+(ad+bc)2
Linear Algebra
4)a) Show that, if A is any n×n matrix with real entries, then there is a n×n symmetric matrix S and a n×n skew symmetric matrix S' such that A=S+S'.
b)Find the solutions to the following system of equations by reducing the corresponding augmented matrix to row-reduced echelon form.
2a+3b+4c+d=8
a+2b+2c+2d=3
a−b+c+3d=3
b)Find the solutions to the following system of equations by reducing the corresponding augmented matrix to row-reduced echelon form.
2a+3b+4c+d=8
a+2b+2c+2d=3
a−b+c+3d=3
Linear Algebra
Let V be the set of all functions that are twice differentiable in R and S={cosx,sinx,xcosx,xsinx}. a)Check that S is a linearly independent set over R.(Hint:Considertheequation a0cosx+a1sinx+a2xcosx+a3xsinx. Put x=0,π,π/ 2 ,π /4 ,etc.and solve for ai.)
b)Let W=[S] and let T:V→V be the function defined by T(f(x))=d2/ dx2(f(x))+2d/ dx(f(x)). Check that T is a linear transformation on V.
c) check that T(W) is subset of W
d) write down the matrix of T on W w.r.t. the basis S
e) Is the matrix of the linear operator T non singular? Justify your answer
b)Let W=[S] and let T:V→V be the function defined by T(f(x))=d2/ dx2(f(x))+2d/ dx(f(x)). Check that T is a linear transformation on V.
c) check that T(W) is subset of W
d) write down the matrix of T on W w.r.t. the basis S
e) Is the matrix of the linear operator T non singular? Justify your answer
Linear Algebra
) Consider the funсtion f:R\{−1}→R defined by f(x)=2x+1 /x+1.
i) Check that f(x) is well defined and 1−1.
2)Check that f(x) is not =2 for any x∈R.
3)Check that g:R\{2}→R given by g(x)=x−1/2-x is well defined and 1-1 Further,check that g(x)=−1for any x∈R.
(4) Check that (f◦g)(x)=x for x∈R\{2} and (g◦f)(x)=x for x∈R\{−1}.
b)Find the direction cosines of the perpendicular from the origin to the plane r·(6i+4j+2√ 3k)+2=0.
i) Check that f(x) is well defined and 1−1.
2)Check that f(x) is not =2 for any x∈R.
3)Check that g:R\{2}→R given by g(x)=x−1/2-x is well defined and 1-1 Further,check that g(x)=−1for any x∈R.
(4) Check that (f◦g)(x)=x for x∈R\{2} and (g◦f)(x)=x for x∈R\{−1}.
b)Find the direction cosines of the perpendicular from the origin to the plane r·(6i+4j+2√ 3k)+2=0.
Linear Algebra
Integrate with respect to x :
∫10xdx
a 4
b 1
c 2
d 1/2
∫10xdx
a 4
b 1
c 2
d 1/2
Linear Algebra
show the matrix [ r1= -9 4 4 ,r2= -8 3 4,r3= -16 8 7,] of order 3*3 is diagonalizable.Obtain the diagonalizable matrix P.
Linear Algebra
What is the difference between Row Echelon form and Reduced Row Echelon form? Also, what is the difference between Gauss Elimination and Gauss Jordon elimination method?
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#333702 on Dec 2022
#333702 on Dec 2022