Linear Algebra

Section I. Solving Linear Systems 15
For instance, the top line says that x =2− 2z +2w. The next section gives a
geometric interpretation that will help us picture the solution sets when they
are written in this way.
2.8 Definition A vector (or column vector) is a matrix with a single column.
A matrix with a single row is a row vector. The entries of a vector are its
components.
Vectors are an exception to the convention of representing matrices with
capital roman letters. We use lower-case roman or greek letters overlined with
an 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.
⎛
�v = ⎝ 1
⎞
3⎠
7
2.9 Definition The linear equation a1x1 + a2x2 + ··· + anxn = d with unknowns
x1,... ,xn is satisfied by
⎛ ⎞
s1
⎜
�s = ⎝ .
⎟
. ⎠
if a1s1 + a2s2 + ··· + ansn = 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, such as the z and w. We
need to define these operations.
2.10 Definition The vector sum of �u and �v is this.
⎛ ⎞ ⎛ ⎞ ⎛
u1 v1
⎜
�u + �v = ⎝ .
⎟ ⎜
. ⎠ + ⎝ .
⎟ ⎜
. ⎠ = ⎝
un
sn
vn
u1 + v1
.
un + vn
In general, two matrices with the same number of rows and the same number
of columns add in this way, entry-by-entry.
2.11 Definition The scalar multiplication of the real number r and the vector
�v is this.
⎛ ⎞ ⎛ ⎞
v1 rv1
⎜
r · �v = r ·
.
⎝ .
⎟ ⎜
. ⎠ =
.
⎝ .
⎟
. ⎠
vn rvn
In general, any matrix is multiplied by a real number in this entry-by-entry way.
⎞
⎟
⎠