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SOLVED: Let T: V-W be a linear map.(a) Define ku (t).(b) Prove that Ker (t) is a subspace of V. Hint: Subspace Criterion.

SOLVED: Let T: V-W be a linear map.(a) Define ku (t).(b) Prove tha…

SOLVED:Suppose V is finite-dimensional. Prove that every linear map on a subspace of V can be extended to a linear map on V. In other words show that if U is a subspace of V and S ∈ℒ(U, W), then there exists T ∈ℒ(V, W) such that T u=S u for all u ∈U

SOLVED:Suppose V is finite-dimensional. Prove that every line…

SOLVED: Linear Algebra2) Let V and W be vector spaces and let f : V → W be a linearmap.a) Let I = y ∈ W | There exist x ∈ V such that f(x) = y. Provethat I is a subspace of W. Note: This subspace I is called theimage of f.b) Let K = x ∈ V | f(x) = 0. Prove that is a subspace of V.Note: The subspace K is called the kernel of f.

SOLVED: Linear Algebra2) Let V and W be vector spaces and let f …

SOLVED: Consider the subspace V span(1,1,1,1)T, (1,0,1,1)T in R4. Let P be the orthogonal projection in R4 that projects vectors onto V. Find the representing matrix for the linear map P with respect to the standard basis. (b) Use your result in (a) to find the projections of the following vectors onto V: v = (1,2,3,4)T and w = (4,3,2,1)T.

SOLVED: Consider the subspace V span(1,1,1,1)T, (1,0,1,1)T in R4. Le…

SOLVED:Let X be a normed linear space and let Y be a dense linear subspace in X. Prove that every y^* ∈Y^* can be extended into an x^*=σy^* of X^*, and that σis an isometric isomorphism of Y^* onto X^*. [Hint: See Theorem 3.8.1.]

SOLVED:Let X be a normed linear space and let Y be a dense linear …

Suppose U is a subspace of a Hilbert space V. Suppose also that W is a Banach space and S: U →W is a bounded linear map. Prove that there exists a bounded linear map T: V →W such that .T|U=S and T=S [If W=𝐅, then this result is just the Hahn-Banach Theorem (6.69) for Hilbert spaces. The result here is stronger because it allows W to be an arbitrary Banach space instead of requiring W to be F. Also, the proof in this Hilbert space context does not require use of Zorn's Lemma or the Axiom of Choi

Suppose U is a subspace of a Hilbert space V. Suppose also tha…

SOLVED: Let V = Span(1, e^x, e^2x) be the subspace of the vector space of all differentiable functions from R to R that is spanned by 1, e^x, and e^2x. (Here, "1" denotes the constant function from R to R whose constant value is 1.)1) Let T: V â†’ V be the differentiation map, so T(f) = (df/dx) for all f âˆˆ V. Note that T is linear (you do not need to prove this). Describe the subspaces im(T) and ker(T); that is, say exactly what these subspaces are as sets.

SOLVED: Let V = Span(1, e^x, e^2x) be the subspace of the vector spa…

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