Linear Algebra, 2020a

Section I. Solving Linear Systems 17
arrow: ⃗a, ⃗b, ... or ⃗α, ⃗β, . . . (boldface is also common: a or α). For instance,
this is a column vector with a third component of 7.
⎛ ⎞
1
⎜ ⎟
⃗v = ⎝3⎠
7
A zero vector is denoted ⃗0. There are many different zero vectors — the one-tall
zero vector, the two-tall zero vector, etc. — but nonetheless we will often say
“the” zero vector, expecting that the size will be clear from the context.
2.9 Definition The linear equation a 1 x 1 + a 2 x 2 + ··· + a n x n = d with unknowns
x 1 ,... ,x n is satisfied by
⎛ ⎞
1...
⎜ ⎟
⃗s = ⎝s
⎠
s n
if a 1 s 1 + a 2 s 2 + ··· + a n s n = d. A vector satisfies a linear system if it satisfies
each equation in the system.
The style of description of solution sets that we use involves adding the
vectors, and also multiplying them by real numbers. Before we give the examples
showing the style we first need to define these operations.
2.10 Definition The vector sum of ⃗u and ⃗v is the vector of the sums.
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
u 1 v 1 u 1 + v 1
⎜ .
⃗u + ⃗v = ⎝
⎟ ⎜ .
. ⎠ + ⎝
⎟ ⎜
. ⎠ = ⎝
⎟
. ⎠
u n v n u n + v n
Note that for the addition to be defined the vectors must have the same
number of entries. This entry-by-entry addition works for any pair of matrices,
not just vectors, provided that they have the same number of rows and columns.
2.11 Definition The scalar multiplication of the real number r and the vector ⃗v
is the vector of the multiples.
⎛ ⎞
v 1
⎜
r · ⃗v = r · ⎝ .
v n
⎟
⎠ =
⎛ ⎞
rv 1
⎜ ⎟
⎝ . ⎠
rv n
As with the addition operation, the entry-by-entry scalar multiplication
operation extends beyond vectors to apply to any matrix.