Linear Algebra, 2020a

96 Chapter Two. Vector Spaces
(b) This set
( ) 0 a
{ | a, b ∈ C and a + b = 0 + 0i}
b 0
1.43 Name a property shared by all of the R n ’s but not listed as a requirement for a
vector space.
1.44 (a) Prove that for any four vectors ⃗v 1 ,...,⃗v 4 ∈ V we can associate their sum
in any way without changing the result.
((⃗v 1 + ⃗v 2 )+⃗v 3 )+⃗v 4 =(⃗v 1 +(⃗v 2 + ⃗v 3 )) + ⃗v 4 =(⃗v 1 + ⃗v 2 )+(⃗v 3 + ⃗v 4 )
= ⃗v 1 +((⃗v 2 + ⃗v 3 )+⃗v 4 )=⃗v 1 +(⃗v 2 +(⃗v 3 + ⃗v 4 ))
This allows us to write ‘⃗v 1 + ⃗v 2 + ⃗v 3 + ⃗v 4 ’ without ambiguity.
(b) Prove that any two ways of associating a sum of any number of vectors give
the same sum. (Hint. Use induction on the number of vectors.)
1.45 Example 1.5 gives a subset of R 2 that is not a vector space, under the obvious
operations, because while it is closed under addition, it is not closed under scalar
multiplication. Consider the set of vectors in the plane whose components have
the same sign or are 0. Show that this set is closed under scalar multiplication but
not addition.
I.2 Subspaces and Spanning Sets
In Example 1.3 we saw a vector space that is a subset of R 2 , a line through the
origin. There, the vector space R 2 contains inside it another vector space, the
line.
2.1 Definition For any vector space, a subspace is a subset that is itself a vector
space, under the inherited operations.
2.2 Example This plane through the origin
⎛ ⎞
x
⎜ ⎟
P = { ⎝y⎠ | x + y + z = 0}
z
is a subspace of R 3 . As required by the definition the plane’s operations are
inherited from the larger space, that is, vectors add in P as they add in R 3
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
x 1 x 2 x 1 + x 2
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎝y 1 ⎠ + ⎝y 2 ⎠ = ⎝y 1 + y 2 ⎠
z 1 z 2 z 1 + z 2