Algebra homework help
Algebra homework help. WORKSHEET 8 (SECTIONS 4.4 AND 4.5)
Let B = {
−1
8
,
1
−7
} and C = {
1
2
,
1
1
} be two basis for R
2
.
(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 =
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 + t
2
, t − 3t
2
, 1 + t − 3t
2}. 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 R
3
.
(1) Use coordinate vectors to show that B is basis for P2.
(2) Find q(t) in P2 such that [q(t)]B =
−1
1
2
.
Determine whether each of the following statements is True or False. Briefly justify your answer.
(a) If B is the standard basis for R
3
then the coordinate vector is itself, that is [x]B = x for all x in
R
3
.
(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