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.

Expert Solution

Colby Messinger answered 2 years ago

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.)

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.

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.

Let V be a finite dimensional vector space over R. If S is a set of...

Let V be a finite dimensional vector space over R. If S is a set of elements in V such that Span(S) = V , what is the relationship between S and the basis of V ?

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!

Let U be a subspace of V . Prove that dim U ⊥ = dim V...

Let U be a subspace of V . Prove that dim U ⊥ = dim V −dim U.

(1) Suppose that V is a vector space and that S = {u,v} is a set...

(1) Suppose that V is a vector space and that S = {u,v} is a set of two vectors in V. Let w=u+v, let x=u+2v, and letT ={w,x} (so thatT is another set of two vectors in V ). (a) Show that if S is linearly independent in V then T is also independent. (Hint: suppose that there is a linear combination of elements of T that is equal to 0. Then ....). (b) Show that if S generates V...

Prove that if U, V and W are vector spaces such that U and V are...

Prove that if U, V and W are vector spaces such that U and V are isomorphic and V and W are isomorphic, then U and W are isomorphic.

V is a subspace of inner-product space R3, generated by vector u =[1 1 2]T and...

V is a subspace of inner-product space R3, generated by vector u =[1 1 2]T and v =[ 2 2 3]T. T is transpose (1) Find its orthogonal complement space V┴ ; (2) Find the dimension of space W = V+ V┴; (3) Find the angle q between u and v; also the angle b between u and normalized x with respect to its 2-norm. (4) Considering v’ = av, a is a scaler, show the angle q’ between u...

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