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- 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.Let A be an mn matrix where mn whose rank is r. a What is the largest value r can be? b How many vectors are in a basis for the row space of A? c How many vectors are in a basis for the column space of A? d Which vector space Rk has the row space as a subspace? e Which vector space Rk has the column space as a subspace?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.
- Spanning the Same Subspace In Exercises 61and 62, show that the sets S1and S2span the same subspace of R3. S1={(0,0,1),(0,1,1),(2,1,1)} S2={(1,1,1),(1,1,2),(2,1,1)}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 bases for the four fundamental subspaces of the matrix. A=[010030101].
- Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.Calculus Repeat Exercise 45 for B={e2x,xe2x,x2e2x}. 45. Calculus Let B={1,x,ex,xex} be a basis for a subspace W of the space of continuous functions, and let Dx be the differential operator on W. Find the matrix for Dx relative to the basis B.Consider the vector spaces P0,P1,P2,...,Pn where Pk is the set of all polynomials of degree less than or equal to k, with standard operations. Show that if jk, then Pj is the subspace of Pk.
