2. Find the four subspaces, including a basis of each, for the matrix (2² 7 -1 -2 4 -5 B= = 1 11 3 3-530 -12

In P,, find the change-of-coordinates matrix from the basis B = {1 + 5t, - 6+t- 29t,1 - 6t} to the standard basis. Then write t as a linear combination of the polynomials in B. find the change-of-coordinates matrix from the basis B to the standard basis. In P2: 1 - 6 1 1 - 6 C-B 5 - 29 (Simplify your answer.) Write t as a linear combination of the polynomials in B. ? =(1+5?) +(-6+t-291) +1 – 6t) %3D (Simplify your answers.)

3. Find an example of a matrix that has Why can't this be a basis for the row space and the mullspace? 1. Explain why the vector v = H and as a basis for its row space and its column space. cannot be a row of A and also in the nullspace of A.

For matrices M and N, the 3rd column of NM is all zeros. Matrix M has 5 columns and none of them are all zero. Prove that the columns of N are linearly dependent.