If W₁ and W₂ are subspace of a vector space V(F), then show that W₁ + W₂ is also a subspace of V(F).
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- Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors {vu,wv,uw} is linearly independent or linearly dependent.Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.
- Find the projection of the vector v=[102]T onto the subspace S=span{[011],[011]}.Repeat Exercise 41 for B={(1,2,2),(1,0,0)} and x=(3,4,4). Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.Find a basis for R2 that includes the vector (2,2).
- Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set {v12v2,2v23v3,3v3v1} linearly dependent or linearly independent? Explain.Prove that in a given vector space V, the zero vector is unique.In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. 34. ,
- Find the bases for the four fundamental subspaces of the matrix. A=[010030101].In Exercises 24-45, use Theorem 6.2 to determine whether W is a subspace of V. V=3, W={[a0a]}Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Prove that the set of all singular 33 matrices is not a vector space.