Linear Algebra, 2020a

34 Chapter One. Linear Systems
(a)
(e)
( ) 1 2
(b)
1 3
⎛ ⎞
2 2 1
⎝ 1 0 5⎠
−1 1 4
( 1
) 2
−3 −6
(c)
( 1 2
) 1
1 3 1
̌ 3.21 Is
(
the given
(
vector
(
in the set generated by the given set?
2 1 1
(a) , { , }
3)
4)
5)
(d)
⎛
1 2
⎞
1
⎝1 1 3⎠
3 4 7
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
−1 2 1
(b) ⎝ 0 ⎠ , { ⎝1⎠ , ⎝0⎠}
1 0 1
⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 1 2 3 4
(c) ⎝3⎠ , { ⎝0⎠ , ⎝1⎠ , ⎝3⎠ , ⎝2⎠}
0 4 5 0 1
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
1 2 3
(d) ⎜0
⎟
⎝1⎠ , { ⎜1
⎟
⎝0⎠ , ⎜0
⎟
⎝0⎠ }
1 1 2
3.22 Prove that any linear system with a nonsingular matrix of coefficients has a
solution, and that the solution is unique.
3.23 In the proof of Lemma 3.6, what happens if there are no non-0 = 0 equations?
̌ 3.24 Prove that if ⃗s and ⃗t satisfy a homogeneous system then so do these vectors.
(a) ⃗s +⃗t (b) 3⃗s (c) k⃗s + m⃗t for k, m ∈ R
What’s wrong with this argument: “These three show that if a homogeneous system
has one solution then it has many solutions — any multiple of a solution is another
solution, and any sum of solutions is a solution also — so there are no homogeneous
systems with exactly one solution.”?
3.25 Prove that if a system with only rational coefficients and constants has a
solution then it has at least one all-rational solution. Must it have infinitely many?