2)Let V₁, V2, W be vector spaces over F. Show that the set Bil(V₁ × V₂, W) ofbilinear maps is a vector space under point-wise addition/scalar multiplication (ie: given f,g bilinear define f + g to be (f + g) (V₁, V₂) := f(V₁, V2) + g(v₁, v₂) and similarly for scalarmultiplication)

Answered: Image /qna-images/answer/05d17851-648e-4f1e-b156-27bb168058d2.jpg

Answered: 2) Let V₁, V2, W be vector spaces over…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.2: Vector Spaces

Problem 37E: Let V be the set of all positive real numbers. Determine whether V is a vector space with the...

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Similar questions

Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Let V be the set of all positive real numbers. Determine whether V is a vector space with the operations shown below. x+y=xyAddition cx=xcScalar multiplication If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
Prove that in a given vector space V, the zero vector is unique.
Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}
Find a basis for R2 that includes the vector (2,2).
Prove that in a given vector space V, the additive inverse of a vector is unique.
Are the vector spaces R4, M2,2, and M1,4 exactly the same? Describe their similarities and differences.
If (x1,y1) + (x2,y2) = (x1x2,y1y2) and c(x1,y1)= (cx1,cy1) are vector spaces, verify if both are a vector space property and identify the properties that it will fail. (Use the 8 theorems)
Find a subset ofbthebvectors v1=(0,2,2,4), v2=(1,0,−1,−3), v3=(2,3,1,1) and v4=(−2,1,3,2)that forms abasis for the space spanned by these vectors. Explain clearly.
2) Let V₁, V2, W be vector spaces over F. Show that the set Bil(V₁ x V₂, W) of bilinear maps is a vector space under point-wise addition/scalar multiplication (ie: given f, g bilinear define f + g to be (f+g)(v₁, 2) = f(v₁, v₂) + g(v₁, v₂) and similarly for scalar multiplication) Solution:
demonstrate whether it's a vector space
3. Let V be the set of ordered pairs (a, b) of real numbers. Show that V is not a vector space over R with addition and scalar multiplication defined by: (i) (a, b) + (c,d) = (a +d, b+c) and k(a, b) = (ka, kb), (ii) (a, b) + (c,d) = (a + c,b+d) and k(a, b) = (a, b), (iii) (a, b) + (c,d) = (0,0) and k(a, b) = (ka, kb), (iv) (a, b) + (c,d) = (ac, bd) and k(a, b) = (ka, kb).

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