Linear Algebra 2270
Homework 2
preparation for the quiz on: 06/04/2015
Axioms of real numbers R:
For arbitrary x, y, z ∈ R we have
RA1) x + y = y + x
RA2) (x + y) + z = y + (x + z)
RA3) there is a natural element of addition denoted by 0, such that for any x ∈ R:
x+0=x
RA4)
RA5)
RA6)
RA7)
there is an additive inverse −x such that x + (−x) = 0
x⋅y =y⋅x
(x ⋅ y) ⋅ z = y ⋅ (x ⋅ z)
there is a natural element of multiplication denoted by 1, such that for any x ∈ R:
x⋅1=x
RA8) If x ≠ 0, there is a multiplicative inverse
RA9) x ⋅ (y + z) = x ⋅ y + x ⋅ z
1
x
such that x ⋅ ( x1 ) = 1
Axioms of vectors Rn :
For arbitrary vectors u, v, w ∈ Rn , we have
V1) u + v = v + u
V2) (u + v) + w = u + (v + w)
V3) there exists a natural element of addition 0 ∈ R, which satisfies u + 0 = u
V4) there exists an additive inverse (−u), such that
u + (−u) = 0
V5) c ⋅ (u + v) = c ⋅ u + c ⋅ v
V6) (c + d) ⋅ u = c ⋅ u + d ⋅ u
V7) c ⋅ (d ⋅ u) = (c ⋅ d)u
V8) 1 ⋅ u = u
We also have:
⎡0⎤
⎢ ⎥
⎢0⎥
⎢ ⎥
0=⎢ ⎥
⎢⋮⎥
⎢ ⎥
⎢0⎥
⎣ ⎦
and if
⎡ u1 ⎤
⎢ ⎥
⎢u ⎥
⎢ ⎥
u = ⎢ 2⎥
⎢ ⋮ ⎥
⎢ ⎥
⎢un ⎥
⎣ ⎦
then
⎡ −u1 ⎤
⎢
⎥
⎢ −u ⎥
⎢ 2⎥
⎥
−u = ⎢
⎢ ⋮ ⎥
⎢
⎥
⎢−un ⎥
⎣
⎦
.
Problems:
1. Determine whether the following matrices are in RREF, REF or neither of them.
1
(1)
(2)
2. Row reduce the following augmented matrices to RREF and then find the solution set of the
corresponding linear system.
3. Determine if linear systems related to the augmented matrices below are consistent or not. If they
are consistent, find the solution set.
4. A system of linear equations with fewer equations than unknowns is sometimes called an undetermined system. Can such a system have a unique solution? Explain.
5. Give an example of a linear system with two equations and three unknowns which is inconsistent.
6. Using axioms RA1)– RA9), prove that if x ∈ R and x ≠ 0, then
1
1
x
=x
7. Using the axioms of the real numbers and the definitions of vector addition and scalar-vector
multiplication prove that V5), V6) and V7) are true.
8. Without knowing the definition of 0 at (1), prove that if V1)-V8) are satisfied, then the natural
element for addition, 0 is unique.
9. Prove that if u ∈ Rn , then (−1) ⋅ u = −u. (There are at least two ways to prove it)
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