Question

Here are some vectors in R 4 : u1 = [1 3 −1 1] u2 = [1 4 −1 1] u3 = [1 0 −1 1] u4 = [2 −1 −2 2] u5 = [1 4 0 1]

(a) Explain why these vectors cannot possibly be independent.

(b) Form a matrix A whose columns are the ui’s and compute the rref(A).

(c) Solve the homogeneous system Ax = 0 in parametric form and then in vector form. (Be sure the clearly identify the row operation used).

(d) Use the vector form solution to write down two relations involving the vectors ui , then solve for the vectors corresponding to the free variables.

(e) Explain why the ui vectors corresponding to the pivot variables form a basis for the subspace U := Span{u1, . . . , u5}

(f) Use the technique in Example 4.94 of the Kuttler textbook to extend the basis found in (e) to a basis for R 4 . (Note: you can use the same row operations as in part (c))

Answer 1

rest you can do easily. Ask if you have any questions. Tgank you