Linear Algebra Worksheet

WORKSHEET 8 (SECTIONS 4.4 AND 4.5)

Let B = { [ −1 8

] ,

[ 1 −7

] } and C = {

[ 1 2

] ,

[ 1 1

] } be two basis for R2.

(1) Suppose [x]B =

[ 2 3

] and [y]C =

[ 2 3

] . Find x, y, are these vectors equal? What does this mean

geometrically? i.e. draw x and y in a plane as a linear combination of vectors in B and C.

(2) Let u =

[ 0 1

] . Find the corresponding coordinate vectors [u]B and [u]C. What does this mean

geometrically?

(3) Find the change of coordinate matrix PB and use PB to compute [u]B from part (2).

WORKSHEET 8 (SECTIONS 4.4 AND 4.5)

Let B = {1 + t2, t − 3t2, 1 + t − 3t2}. Note that any question/property that we can ask about these polynomials in P2 translates into the same question/property about their corresponding coordinate vectors in R3.

(1) Use coordinate vectors to show that B is basis for P2.

(2) Find q(t) in P2 such that [q(t)]B =

 −11

2

 .

Determine whether each of the following statements is True or False. Briefly justify your answer.

(a) If B is the standard basis for R3 then the coordinate vector is itself, that is [x]B = x for all x in R3.

(b) If there exists a set of 3 vectors that spans a vector space V then dim V = 3.

(c) If there exists a linearly independent set of 3 vectors in V then dim V ≥ 3.

(d) If dim V = 3 then every set of 2 nonzero vectors in V is linearly independent.

(e) If dim V = 3 then any set of 4 vectors spans V .