Table of Contents
Linear Combination
Definition. (Linear combination) Given vectors \(\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v}_k\) in \(\mathbb{R}^n\) and given scalars \(c_1,c_2,\dots,c_k\), the vector \(\mathbf{w}\) defined as
$$
\mathbf{w}=c_1\mathbf{v}_1+c_2\mathbf{v}_2+\cdots+c_k\mathbf{v}_k
$$
is called a linear combination of \(\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v}_k\)
the homogeneous equation
only has the trivial solution. Otherwise, the set is called linearly dependent.
EXAMPLE 1. The set
The vector \(\mathbf{v}=\begin{bmatrix}7\\7\\15\end{bmatrix}\) is a linear combination of the vectors \(\mathbf{v}_1=\begin{bmatrix}2\\3\\4\end{bmatrix}\) and \(\mathbf{v}_2=\begin{bmatrix}-1\\2\\-3\end{bmatrix}\) since
\[\mathbf{v} = 3\mathbf{v}_1 -1 \mathbf{v}_2.\]
In this case \(c_1=3\) and \(c_2=-1\).
Linear Independence
Definition. (Linear independence) A set of vectors \(\lbrace\mathbf{v}_1,\mathbf{v}_2,\dots,\mathbf{v}_n\rbrace\) in \(\mathbb{R}^n\) is linearly independent if the homogeneous equation
$$
x_1\mathbf{v}_1+x_2\mathbf{v}_2+\cdots+x_n\mathbf{v}_n = \mathbf{0}
$$
only has the trivial solution. Otherwise, the set is called linearly dependent.
EXAMPLE 2. The set
\(\left\lbrace \begin{bmatrix}2\\0\\-1\end{bmatrix},\begin{bmatrix}-1\\0\\-1\end{bmatrix},\begin{bmatrix}1\\1\\1\end{bmatrix} \right\rbrace\) is linear independent. We need to check that the system
\[\begin{align*}
2x_1-x_2+x_3&=0\\
x_3&=0\\
-x_1-x_2+x_3&=0
\end{align*}\]
has only a trivial solution. Indeed, the second equation provides that \(x_3=0\). Substituting \(x_3=0\) into the third and first equation, and isolating \(x_1\) on the third we have that \(x_1=-x_2\). Substituting this into the first equation we obtain that \(x_1=0\) and, from the previous relation, \(x_2=0\). So the system has a trivial solution.