Linear combinations
Linear combinations of vectors and spans.
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Study Linear combinations
Linear combination
Definition 1. Let \(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_k\in \mathbb{R}^n\) and \(\lambda_1,\lambda_2,\ldots,\lambda_k\in\mathbb{R}\) be given. Then we call the vector \[\mathbf{v} = \lambda_1\mathbf{v}_1+\lambda_2\mathbf{v}_2+\ldots+\lambda_k\mathbf{v}_k\] a linear combination of \(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_k\). We also say that \(\mathbf{v}\) can be generated by \(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_k\).
Example 2.
\(5\cdot \begin{pmatrix} 3\\1\\2 \end{pmatrix} + 7\cdot \begin{pmatrix} -1\\0\\2 \end{pmatrix}\) is a linear combination of \(\begin{pmatrix} 3\\1\\2 \end{pmatrix}\) and \(\begin{pmatrix} -1\\0\\2 \end{pmatrix}\).
\(\begin{pmatrix} 3\\-7\\-1 \end{pmatrix} = \frac{3}{2}\cdot \begin{pmatrix} 2\\0\\0 \end{pmatrix} -2\cdot \begin{pmatrix} 1\\3\\0 \end{pmatrix} +\frac{1}{2}\cdot \begin{pmatrix} 4\\-2\\-2 \end{pmatrix}\) can be written as the linear combination of \(\begin{pmatrix} 2\\0\\0 \end{pmatrix}, \begin{pmatrix} 1\\3\\0 \end{pmatrix}, \begin{pmatrix} 4\\-2\\-2 \end{pmatrix}\).
Example 3. Every point (position vector) \(\mathbf{x}\in\mathbb{R}^3\) lying in the \(x_1x_2\)-plane can be generated by the vectors \[\mathbf{e}_1:=\begin{pmatrix} 1\\0\\0 \end{pmatrix},~ \mathbf{e}_2:=\begin{pmatrix} 0\\1\\0 \end{pmatrix}\, .\]
Example 4. Every point (position vector) \(\mathbf{x}\in\mathbb{R}^2\) can be generated by the vectors \[\mathbf{v}_1:=\begin{pmatrix} 2\\1 \end{pmatrix},~ \mathbf{v}_2:=\begin{pmatrix} -1\\1 \end{pmatrix}\, .\]
Span
Definition 5. Consider vectors \(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_k\in \mathbb{R}^n\). The set of all linear combinations of the vectors \(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_k\in \mathbb{R}^n\) is called span (or linear hull) of \(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_k\) . Shortly: \[\text{Span}(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_k):=\left\{\lambda_1\mathbf{v}_1+\lambda_2\mathbf{v}_2+\ldots+\lambda_k\mathbf{v}_k:~ \lambda_1,\ldots,\lambda_k\in\mathbb{R}\right\}\, .\]
Exercise 6. Sketch the following spans.
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