Linear Algebra, 2020a

Section III. Basis and Dimension 127
Exercises
̌ 1.20 Decide if each is a basis for P 2 .
(a) 〈x 2 − x + 1, 2x + 1, 2x − 1〉 (b) 〈x + x 2 ,x− x 2 〉
̌ 1.21 Decide ⎛ ⎞if ⎛each ⎞is⎛
a basis ⎞ for R 3 ⎛.
⎞ ⎛ ⎞
1 3 0
1 3
⎛
(a) 〈 ⎝2⎠ , ⎝2⎠ , ⎝0⎠〉 (b) 〈 ⎝2⎠ , ⎝2⎠〉 (c) 〈 ⎝
3 1 1
3 1
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
0 1 1
(d) 〈 ⎝ 2 ⎠ , ⎝1⎠ , ⎝3⎠〉
−1 1 0
0
2
−1
̌ 1.22 Represent
(
the
(
vector
(
with
)
respect to the basis.
1 1 −1
(a) , B = 〈 , 〉⊆R
2)
1)
2
1
(b) x 2 + x 3 , D = 〈1, 1 + x, 1 + x + x 2 ,1+ x + x 2 + x 3 〉⊆P 3
⎛ ⎞
0
(c) ⎜−1
⎟
⎝ 0 ⎠ , E 4 ⊆ R 4
1
⎞ ⎛ ⎞ ⎛ ⎞
1 2
⎠ , ⎝1⎠ , ⎝5⎠〉
1 0
1.23 Represent the vector with respect to each of the two bases.
( ) ( ) ( ( ( 3
1 1 1 1
⃗v = B 1 = 〈 , 〉, B 2 = 〈 , 〉
−1
−1 1)
2)
3)
1.24 Find a basis for P 2 , the space of all quadratic polynomials. Must any such
basis contain a polynomial of each degree: degree zero, degree one, and degree two?
1.25 Find a basis for the solution set of this system.
x 1 − 4x 2 + 3x 3 − x 4 = 0
2x 1 − 8x 2 + 6x 3 − 2x 4 = 0
̌ 1.26 Find a basis for M 2×2 , the space of 2×2 matrices.
̌ 1.27 Find a basis for each.
(a) The subspace {a 2 x 2 + a 1 x + a 0 | a 2 − 2a 1 = a 0 } of P 2
(b) The space of three-wide row vectors whose first and second components add
to zero
(c) This subspace of the 2×2 matrices
( ) a b
{ | c − 2b = 0}
0 c
1.28 Find a basis for each space, and verify that it is a basis.
(a) The subspace M = {a + bx + cx 2 + dx 3 | a − 2b + c − d = 0} of P 3 .
(b) This subspace of M 2×2 .
( ) a b
W = { | a − c = 0}
c d
1.29 Check Example 1.6.
̌ 1.30 Find the span of each set and then find a basis for that span.