Question

1) Your colleague on the skyscraper design team brings in a quiver (a set) full of
vectors. She would like to know which ones could be used as a basis. Select vectors from the
quiver that will span the subspace represented by all vectors in the quiver.
Quiver Set: v1 = (1,−1,5,2), v2 = (−2,3,1,0), v3 = (4,−5,9,4),

v4 = (0,4,2,−3), v5 =(−7,18,2,−8)

Part 2: She wants to know if they span R6. What is the problem with her question? What is the most dimensions that such vectors could span? How many dimensions does in fact this set span? Explain.

Part 3: Her boss brought in a stray vector she found in the parking lot. Does it belong in the quiver? (That is, does it belong to the space spanned by the basis vectors?) If so give her the vector’s coordinates w.r.t. the basis. {Even though you found a basis in problem 1, start with these as your basis instead: u1 = (0,2,1,0), u2 = (1,−1,0,0), u3 = (1,2,0,−1) }


stray vector = (3,-11,11,12)

Part 4: Here she comes again. “Would you please convert my quiver’s basis in to an orthogonal basis set?” {Start with the basis used to start off problem 3.}

Part 5: “And, ah, could you please make that an orthonormal set?

Part 6: After a while your colleague gets curious about the space that cannot be ‘reached’ by her quiver’s basis. (Who wouldn’t be.) This is the space between full R4 and the subspace represented by the quiver. Find a set of vectors that span this ‘in-between’ space (i.e. find a basis for the orthogonal complement to the subspace of the quiver’s basis — null space anyone?). {Once again use the following as your basis vectors instead of those found earlier: v1 = (1,4,5,2), v2 = (2,1,3,0), v3 = (−1,3,2,2). Don’t forget to make these ROWs of a matrix then finding the matrix’s null space!}.

Part 7: Finally, she needs to share her building’s vectors with the city inspector, who uses the R3 standard basis aligned with East-West/North-South/Sky-Earth. She had reengineered all her vectors in terms of R3 using the basis B (below). She asks you “Can you please create a transition matrix that converts my vectors into their standard basis vectors?” Find the transition matrix PB→S

B = {(1,2,1), (2,5,0), (3, 3, 8)}, S = {(1,0,0), (0,1,0), (0,0,1)}

Answer 1