On the relationship between linear combinations of vectors and systems of linear equations.
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Linear
combination \(\rightarrow\) system of
linear equations
The following question appears frequently in Linear Algebra:
"Given a fixed vector \(\mathbf{v}\),
can it be written as linear combination of some other given vectors
\(\mathbf{v}_1,\mathbf{v}_2,\ldots\)?"
\[\begin{pmatrix}
8\\17\\0
\end{pmatrix}
=
\mathbf{x_1}
\begin{pmatrix}
1\\2\\-1
\end{pmatrix}
+
\mathbf{x_2}
\begin{pmatrix}
-3\\-5\\5
\end{pmatrix}
+
\mathbf{x_3}
\begin{pmatrix}
5\\9\\-4
\end{pmatrix}\] This corresponds to asking whether a certain
system of linear equations has a solution: \[\begin{array}{rrrrrrr}
8 & = & 1x_1 & - & 3x_2 & + & 5x_3 \\
17 & = & 2x_1 & - & 5x_2 & + & 9x_3 \\
0 & = & -1x_1 & + & 5x_2 & - & 4x_3
\end{array}\]
System of linear
equations
Definition 1. Let \(m,n\in\mathbb{N}\). A system of
linear equations (LES) in the variables \(x_1,x_2,\ldots,x_n\) is of the form \[\begin{aligned}
a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n & = b_1\\
a_{21}x_1 + a_{22}x_2 + \ldots + a_{2n}x_n & = b_2\\
& \vdots\\
a_{m1}x_1 + a_{m2}x_2 + \ldots + a_{mn}x_n & = b_m ~
\end{aligned}\]
with \(a_{ij}\) and \(b_i\) being (usually real) numbers. An
assignment of values for \(x_1,\ldots,x_n\) such that all equations
are satisfied is called a solution of this system of
equations. Such a solution is written as a vector.
Example 2. The system \[\begin{array}{rrrrrrr}
1x_1 & - & 3x_2 & + & 5x_3 & = & 8\\
2x_1 & - & 5x_2 & + & 9x_3 & = &17 \\
-1x_1 & + & 5x_2 & - & 4x_3 & = & 0
\end{array}\] is a system of linear equations with 3 equations
and 3 variables. A solution is \[\begin{pmatrix}
x_1\\x_2\\x_3
\end{pmatrix}
=
\begin{pmatrix}
7\\3\\2
\end{pmatrix}\, .\]
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