6. Turn the basis below of the three dimensional subspace U CR¹ into an orthogonal basis, through Gram-Schmidt via Gaussian elimination; show tableau work. 2 0 2 U = span[b₁ = b₂ == , b3 := 1 1 6 0 0

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 the vector space of all 33 diagonal matrices. What is the dimension of this vector space?

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.