A First Course in Linear Algebra, 2017a

9.1. Algebraic Considerations 451
Example 9.4: R n
R n , under the usual operations of vector addition and scalar multiplication, is a vector space.
Solution. To show that R n is a vector space, we need to show that the above axioms hold. Let ⃗x,⃗y,⃗z be
vectors in R n . We first prove the axioms for vector addition.
• To show that R n is closed under addition, we must show that for two vectors in R n their sum is also
in R n .Thesum⃗x +⃗y is given by:
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
x 1 y 1 x 1 + y 1
x 2
⎢ ⎥
⎣ . ⎦ + y 2
⎢ ⎥
⎣ . ⎦ = x 2 + y 2
⎢ ⎥
⎣ . ⎦
x n y n x n + y n
The sum is a vector with n entries, showing that it is in R n . Hence R n is closed under vector addition.
• To show that addition is commutative, consider the following:
⎡ ⎤ ⎡ ⎤
x 1 y 1
x 2
⃗x +⃗y = ⎢ ⎥
⎣ . ⎦ + y 2
⎢ ⎥
⎣ . ⎦
x n y n
=
⎡
⎢
⎣
Hence addition of vectors in R n is commutative.
⎤
x 1 + y 1
x 2 + y 2
⎥
. ⎦
x n + y n
⎡ ⎤
y 1 + x 1
y 2 + x 2
= ⎢ ⎥
⎣ . ⎦
y n + x n
⎡ ⎤ ⎡ ⎤
y 1 x 1
y 2
= ⎢ ⎥
⎣ . ⎦ + x 2
⎢ ⎥
⎣ . ⎦
y n x n
= ⃗y +⃗x
• We will show that addition of vectors in R n is associative in a similar way.
⎛⎡
⎤ ⎡ ⎤⎞
⎡ ⎤
x 1 y 1 z 1
x 2
(⃗x +⃗y)+⃗z = ⎜⎢
⎥
⎝⎣
. ⎦ + y 2
⎢ ⎥⎟
⎣ . ⎦⎠ + z 2 ⎢ ⎥
⎣ . ⎦
x n y n z n