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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 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]}.
- Find a basis for the vector space of all 33 diagonal matrices. What is the dimension of this vector space?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.How would I find whether there is a linear algebra subspace in R^3, inclusion of zero vector, closure under vector addition or scalar multiplication?
- Show that [0,1] and (0,1] as subspaces of R with the usual topology are not homeomorphic.Find an orthogonal projection of (6, 0, 0) (1) along the vector (1, 2, 1)(2) On the plane determined by (1, 2, 1) and (1, −1, 1)(3) On the subspace spanned by (1, 2, 1), (1, −1, 1) and (3, 0, −3)Why are [0,1] and (0,1] which are subspaces of the real number system with the usual topology not homeomorphic?