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- Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}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.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.
- Prove that in a given vector space V, the zero vector is unique.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.Does the set of all functions that vanish at x = 0 and x = L form a vector space? If it does, explicitly show that it satisfies all eight properties required of a vector space. If not, which property fails? Show how it fails.
- Suppose V1, ..., Vn are vector spaces then does L(V1 x...x Vn, U) have the same dim with L(V1, U) x...x L(Vn, U), since these L(V1 x...x Vn, U) and L(V1, U) x...x L(Vn, U) are both vector spaces.Show that Green's Theorem holds for the vector fieldF(x, y)=(x +y, 3x) at a square of side 2.Let F(x,y,z)=(−6xz2,6xyz,6xy3z)F(x,y,z)=(−6xz2,6xyz,6xy3z) be a vector field and f(x,y,z)=x3y2zf(x,y,z)=x3y2z.∇f=(∇f=( , , )).∇×F=(∇×F=( , , )).F×∇f=(F×∇f=( , , )).F⋅∇f=F⋅∇f= .
- Prove that in n-dimensional space, symmetric and skew-symmetric tensor have ( 1) 2n nand ( 1) 2n nindependent components respectively.Show that the vector field F = ( -z, 0, x) is orthogonal to the position ----+ vector OP at each point P. Give an example of another vector field with this property.Evaluate the 1-form φ = x2yz dx − z dy on the vector field V (x, y, z) = xU1(x, y, z) − yzU2(x, y, z) + 6zU3(x, y, z)..
