13. If R4 is given the standard inner product, find an orthonormal basis for the subspace S = {(x1, x2, x3, x4) basis for all R4. R4 x1 + x2 + x3 + x4 = 0}, and extend this basis to an orthonormal :

Find a basis for the vector space of all 33 diagonal matrices. What is the dimension of this vector space?

Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=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.