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Linear Algebra
Let M2(R) be the vector space of all 2*2 matrices over R . Prove that M2(R)=S+T , where S={A belongs to M2(R)|aii=0 , i=1,2} and T={B belongs to M2(R)|bij=0 , i does not equal to the j } are subspaces of M2(R).
Linear Algebra
Find a basis of the subspace of R3 of the solution of the equation X+Y+Z=0 .
Linear Algebra
Determine whether the following subsets are subspaces of the given vector spaces:
(1) S={A belongs to Mn(C) | A is skew-Hermition} , vector space Mn(C) over C.
(2) W={ f(x) belongs F[x] f(0)=0 }, vector space F[x] over F.
(1) S={A belongs to Mn(C) | A is skew-Hermition} , vector space Mn(C) over C.
(2) W={ f(x) belongs F[x] f(0)=0 }, vector space F[x] over F.
Linear Algebra
Is there any vector space V, such that for no two subspaces U and W
V = U ⊕ W
Give proper reasoning.?
V = U ⊕ W
Give proper reasoning.?
Linear Algebra
Let (V,<,>) be an inner product space over C and T belongs to A(V) . Prove that if <Tx , Ty>=<x,y> for all x,y belongs to V , then T is unitary.
Linear Algebra
Find the rank of the quadratic form 2x2+2y2+3z2-xy+4yz +5xz .
Linear Algebra
Find the dual basis of the basis e1=(1,1,2) , e2=(1,0,1) , e3=(2,1,0) of the vector space R3 over R.
Linear Algebra
Obtain an orthogonal basis with respect to the standard inner product for the subspace of R4 defined by { (x,y,w,z)| x+2y+z+3w=0 , x-y-z=0}.
Linear Algebra
Find the normal cononial form of the quadratic form 2xy+2yz-x2-y2-z2 . Hence compute its signature.
Linear Algebra
find out basis vector for V= matrix of 2×2 of complex numbers(C) over the field of real numbers(R) . what is dimension of matrix of 2×2 of complex numbers(C) over real numbers(R)
