Linear Algebra, 2020a

128 Chapter Two. Vector Spaces
(a) {1 + x, 1 + 2x} in P 2 (b) {2 − 2x, 3 + 4x 2 } in P 2
̌ 1.31 Find a basis for each of these subspaces of the space P 3 of cubic polynomials.
(a) The subspace of cubic polynomials p(x) such that p(7) =0
(b) The subspace of polynomials p(x) such that p(7) =0 and p(5) =0
(c) The subspace of polynomials p(x) such that p(7) =0, p(5) =0, and p(3) =0
(d) The space of polynomials p(x) such that p(7) = 0, p(5) = 0, p(3) = 0,
and p(1) =0
1.32 We’ve seen that the result of reordering a basis can be another basis. Must it
be?
1.33 Can a basis contain a zero vector?
̌ 1.34 Let 〈⃗β 1 , ⃗β 2 , ⃗β 3 〉 be a basis for a vector space.
(a) Show that 〈c 1
⃗β 1 ,c 2
⃗β 2 ,c 3
⃗β 3 〉 is a basis when c 1 ,c 2 ,c 3 ≠ 0. What happens
when at least one c i is 0?
(b) Prove that 〈⃗α 1 , ⃗α 2 , ⃗α 3 〉 is a basis where ⃗α i = ⃗β 1 + ⃗β i .
1.35 Find one vector ⃗v that will make each into a basis for the space.
⎛ ⎞ ⎛ ⎞
( 1 0
1
1)
(a) 〈
,⃗v〉 in R 2 (b) 〈 ⎝ ⎠ , ⎝ ⎠ ,⃗v〉 in R 3 (c) 〈x, 1 + x 2 ,⃗v〉 in P 2
1
0
1
0
̌ 1.36 Consider 2 + 4x 2 ,1+ 3x 2 ,1+ 5x 2 ∈ P 2 .
(a) Find a linear relationship among the three.
(b) Represent them with respect to B = 〈1 − x, 1 + x, x 2 〉.
(c) Check that the same linear relationship holds among the representations, as
in Lemma 1.18.
̌ 1.37 Where 〈⃗β 1 ,...,⃗β n 〉 is a basis, show that in this equation
c 1
⃗β 1 + ···+ c k
⃗β k = c k+1
⃗β k+1 + ···+ c n
⃗β n
each of the c i ’s is zero. Generalize.
1.38 A basis contains some of the vectors from a vector space; can it contain them
all?
1.39 Theorem 1.12 shows that, with respect to a basis, every linear combination is
unique. If a subset is not a basis, can linear combinations be not unique? If so,
must they be?
1.40 A square matrix is symmetric if for all indices i and j, entryi, j equals entry
j, i.
(a) Find a basis for the vector space of symmetric 2×2 matrices.
(b) Find a basis for the space of symmetric 3×3 matrices.
(c) Find a basis for the space of symmetric n×n matrices.
1.41 We can show that every basis for R 3 contains the same number of vectors.
(a) Show that no linearly independent subset of R 3 contains more than three
vectors.
(b) Show that no spanning subset of R 3 contains fewer than three vectors. Hint:
recall how to calculate the span of a set and show that this method cannot yield
all of R 3 when we apply it to fewer than three vectors.