Consider the following set of vectors in R6 S={[-9 7 -8 3 0 -5], [1 -7...

Consider the following set of vectors in R6

S={[-9 7 -8 3 0 -5], [1 -7 3 2 -8 -8], [-6 -14 1 9 -23 -29], [11 -21 14 1 -16 -11], [8 16 -8 8 10 1], [17 -7 13 -8 8 18]

(a) (2 points) Demonstrate that S is not a basis for R6.

(b) (4 points) Let H = Span S. Find a basis for H and determine its dimension.

(c) (2 points) Determine whether v= [1,1,1,−1,−1,−1] ^T belongs in H or not.

(d) (6 points) Find a basis for R6 consists of the basis vectors of H found in part (b) and some additional linearly independent vectors.

(e) (4 points)Suppose A is a 6×6 matrix and T(x) =Ax. Show that T(H), the set of images of vectors of H, is a subspace of R6.

(f) (3 points) Show that dimT(H)≤dimH

(g) (4 points) Suppose A is invertible. Show that dimT(H) = dimH.

(h) (5 points)Suppose K is a 4 dimensional subspace of R6. Show that Hand K must have a nonzero vector in common.Hint: Start with bases for the two subspaces. If H andKonly have the trivial vector in common, then what’s a basis for the subspace H+K?

Expert Solution

Colby Messinger answered 1 year ago

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