Suppose V is a ﬁnite dimensional inner product space. Prove that every orthogonal operator on V...

Suppose V is a ﬁnite dimensional inner product space. Prove that every orthogonal operator on V , i.e., <T(u),T(v)> = <u,v>, ∀u,v ∈ V , is an isomorphism.

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Colby Messinger answered 1 year ago

Let T be a linear operator on a finite-dimensional complex vector space V . Prove that...

Let T be a linear operator on a finite-dimensional complex vector space V . Prove that T is diagonalizable if and only if for every λ ∈ C, we have N(T − λIV ) = N((T − λIV )2).

Let T be an operator on a finite-dimensional complex vector space V, and suppose that dim...

Let T be an operator on a finite-dimensional complex vector space V, and suppose that dim Null T = 3, dimNullT2 =6. Prove that T does not have a square root; i.e. there does not exist any S ∈ L (V) such that S2 = T.

Suppose U is a subspace of a finite dimensional vector space V. Prove there exists an...

Suppose U is a subspace of a finite dimensional vector space V. Prove there exists an operator T on V such that the null space of T is U.

Let V be a finite-dimensional vector space over C and T in L(V). Prove that the...

Let V be a finite-dimensional vector space over C and T in L(V). Prove that the set of zeros of the minimal polynomial of T is exactly the same as the set of the eigenvalues of T.

suppose that T : V → V is a linear map on a finite-dimensional vector space...

suppose that T : V → V is a linear map on a finite-dimensional vector space V such that dim range T = dim range T2. Show that V = range T ⊕null T. (Hint: Show that null T = null T2, null T ∩ range T = {0}, and apply the fundamental theorem of linear maps.)

Suppose V is finite-dimensional and S, T are operators on V . Prove that ST is...

Suppose V is finite-dimensional and S, T are operators on V . Prove that ST is bijective if and only if S and T are both bijective. Note: Don’t forget that bijective maps are precisely those that have an inverse!

V and W are finite dimensional inner product spaces,T: V→W is a linear map 1A: Give...

V and W are finite dimensional inner product spaces,T: V→W is a linear map 1A: Give an example of a map T from R2 to itself (with the usual inner product) such that〈Tv,v〉= 0 for every map. 1B: Suppose that V is a complex space. Show that〈Tu,w〉=(1/4)(〈T(u+w),u+w〉−〈T(u−w),u−w〉)+(1/4)i(〈T(u+iw),u+iw〉−〈T(u−iw),u−iw〉 1C: Suppose T is a linear operator on a complex space such that〈Tv,v〉= 0 for all v. Show that T= 0 (i.e. that Tv=0 for all v).

Problem 3. An isometry between inner-product spaces V and W is a linear operator L in...

Problem 3. An isometry between inner-product spaces V and W is a linear operator L in B (V ,W) that preserves norms and inner-products. If x, y in V and if L is an isometry, then we have <L(x),L(y)>_W = <x, y>_V . Suppose that V and W are both real, n-dimensional inner-product spaces. Thus the scalar field for both is R and both of them have a basis consisting of n elements. Show that V and W are isometric...

Suppose a 2-dimensional space is spanned by the coordinates (v, x) and that the line element...

Suppose a 2-dimensional space is spanned by the coordinates (v, x) and that the line element is defined by: ds2 = -xdv2 + 2dvdx i) Assuming that the nature and properties of the spacetime line element still hold, show that a particle in the negative x-axis can never venture into the positive x-axis. That is, show that a particle becomes trapped if it travels into the negative x-axis. ii) Draw a representative light cone that illustrates how a particle is...

this is abstract algebra Let M be a Q[x]-module which is ﬁnite-dimensional as a vector space....

this is abstract algebra Let M be a Q[x]-module which is ﬁnite-dimensional as a vector space. What is its torsion submodule?

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