(3) Let V be a vector space over a field F. Suppose that a ? F,...

(3) Let V be a vector space over a field F. Suppose that a ? F, v ? V and av = 0. Prove that a = 0 or v = 0.

(4) Prove that for any field F, F is a vector space over F.

(5) Prove that the set V = {a0 + a1x + a2x 2 + a3x 3 | a0, a1, a2, a3 ? R} of polynomials of degree ? 3 is a vector space over R (with respect to the usual addition and scalar multiplication of polynomials).

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milcah answered 1 year ago

Proof: Let S ⊆ V be a subset of a vector space V over F. We...

Proof: Let S ⊆ V be a subset of a vector space V over F. We have that S is linearly dependent if and only if there exist vectors v1, v2, . . . , vn ∈ S such that vi is a linear combination of v1, v2, . . . , vi−1, vi+1, . . . , vn for some 1 ≤ i ≤ n.

Let V be a vector space of dimension 1 over a field k and choose a...

Let V be a vector space of dimension 1 over a field k and choose a fixed nonzero element voe V, which is therefore a basis. Let W be any vector space over k and let woe W be an arbitrary vector. Show that there is a unique linear transformation T: V → W such that T(v)= wo. [Hint: What must T(Avo) be?)

Let V and W be finite dimensional vector spaces over a field F with dimF(V )...

Let V and W be finite dimensional vector spaces over a field F with dimF(V ) = dimF(W ) and let T : V → W be a linear map. Prove there exists an ordered basis A for V and an ordered basis B for W such that [T ]AB is a diagonal matrix where every entry along the diagonal is either a 0 or a 1.

Please show work and explain Prove that a vector space V over a field F is...

*Please show work and explain* Prove that a vector space V over a field F is isomorphic to the vector space L(F,V) of all linear maps from F to V.

Question 1. Let V and W be finite dimensional vector spaces over a field F with...

Question 1. Let V and W be finite dimensional vector spaces over a field F with dimF(V ) = dimF(W) and let T : V → W be a linear map. Prove there exists an ordered basis A for V and an ordered basis B for W such that [T] A B is a diagonal matrix where every entry along the diagonal is either a 0 or a 1. Hint 1. Suppose A = {~v1, . . . , ~vn}...

(10pt) Let V and W be a vector space over R. Show that V × W...

(10pt) Let V and W be a vector space over R. Show that V × W together with (v0,w0)+(v1,w1)=(v0 +v1,w0 +w1) for v0,v1 ∈V, w0,w1 ∈W and λ·(v,w)=(λ·v,λ·w) for λ∈R, v∈V, w∈W is a vector space over R. (5pt)LetV beavectorspaceoverR,λ,μ∈R,andu,v∈V. Provethat (λ+μ)(u+v) = ((λu+λv)+μu)+μv. (In your proof, carefully refer which axioms of a vector space you use for every equality. Use brackets and refer to Axiom 2 if and when you change them.)

Let V be the vector space of all functions f : R → R. Consider the...

Let V be the vector space of all functions f : R → R. Consider the subspace W spanned by {sin(x), cos(x), e^x , e^−x}. The function T : W → W given by taking the derivative is a linear transformation a) B = {sin(x), cos(x), e^x , e^−x} is a basis for W. Find the matrix for T relative to B. b)Find all the eigenvalues of the matrix you found in the previous part and describe their eigenvectors. (One...

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(3) Let V be a vector space over a field F. Suppose that a ? F,...

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