Linear Algebra, 2020a

104 Chapter Two. Vector Spaces
(c) The superhero and the villain are transported to a three dimensional space
where the superhero now has three devices.
⎛ ⎞
⎛ ⎞
⎛ ⎞
−1
2
5
hoverboard: ⎝ 0 ⎠ magic carpet: ⎝1⎠
scooter: ⎝ 4 ⎠
3
0
−9
Is there anywhere that the villain could safely hide? If so, give one such location
and if not, explain why not.
2.25 Which of these are members of the span [{cos 2 x, sin 2 x}] in the vector space of
real-valued functions of one real variable?
(a) f(x) =1 (b) f(x) =3 + x 2 (c) f(x) =sin x (d) f(x) =cos(2x)
̌ 2.26 Which of these sets spans R 3 ? That is, which of these sets has the property
that any three-tall vector can be expressed as a suitable linear combination of the
set’s elements? ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 0 0
2 1 0
1 3
(a) { ⎝0⎠ , ⎝2⎠ , ⎝0⎠} (b) { ⎝0⎠ , ⎝1⎠ , ⎝0⎠} (c) { ⎝1⎠ , ⎝0⎠}
0 0 3
1 0 1
0 0
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 3 −1 2
2 3 5 6
(d) { ⎝0⎠ , ⎝1⎠ , ⎝ 0 ⎠ , ⎝1⎠} (e) { ⎝1⎠ , ⎝0⎠ , ⎝1⎠ , ⎝0⎠}
1 0 0 5
1 1 2 2
̌ 2.27 Parametrize each subspace’s description. Then express each subspace as a
span.
(a) The subset {(a b c) | a − c = 0} of the three-wide row vectors
(b) This subset of M 2×2
( ) a b
{ | a + d = 0}
c d
(c) This subset of M 2×2
( ) a b
{ | 2a − c − d = 0 and a + 3b = 0}
c d
(d) The subset {a + bx + cx 3 | a − 2b + c = 0} of P 3
(e) The subset of P 2 of quadratic polynomials p such that p(7) =0
̌ 2.28 Find a set to span the given subspace of the given space. (Hint. Parametrize
each.)
(a) the xz-plane in R
⎛ ⎞
3
x
(b) { ⎝y⎠ | 3x + 2y + z = 0} in R 3
z
⎛ ⎞
x
(c) { ⎜y
⎟
⎝ z ⎠ | 2x + y + w = 0 and y + 2z = 0} in R4
w
(d) {a 0 + a 1 x + a 2 x 2 + a 3 x 3 | a 0 + a 1 = 0 and a 2 − a 3 = 0} in P 3
(e) The set P 4 in the space P 4
(f) M 2×2 in M 2×2
2.29 Is R 2 a subspace of R 3 ?