Linear Algebra – Systems of linear equations – Vector equations

Notations for vectors in \(\mathbb{R}^n\) are: \(\mathbf{u}=\begin{pmatrix}u_1\\u_2\\\vdots\\u_n\end{pmatrix},\;\mathbf{v}=\begin{pmatrix}v_1\\v_2\\\vdots\\v_n\end{pmatrix} \;\Longrightarrow\;c\mathbf{u}+d\mathbf{v}=\begin{pmatrix}cu_1+dv_1\\cu_2+dv_2\\\vdots\\cu_n+dv_n\end{pmatrix}\) for \(c,d\in\mathbb{R}\) and \(\mathbf{0}=\begin{pmatrix}0\\0\\\vdots\\0\end{pmatrix}\) is called the zero vector.

If \(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_p\) are vectors in \(\mathbb{R}^n\) and \(c_1,c_2,\ldots,c_p\) are scalars in \(\mathbb{R}\), then \(c_1\mathbf{v}_1+c_2\mathbf{v}_2+\cdots+c_p\mathbf{v}_p\) is called a linear combination of the vectors \(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_p\). The scalars \(c_1,c_2,\ldots,c_p\) are called the weights of the linear combination.

Theorem: Let \(\mathbf{u}\), \(\mathbf{v}\), \(\mathbf{w}\) and \(\mathbf{0}\) be vectors in \(\mathbf{R}^n\) and \(c\) and \(d\) scalars in \(\mathbb{R}\), then we have:

\(\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}\) \((\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})\) \(\mathbf{u}+\mathbf{0}=\mathbf{0}+\mathbf{u}=\mathbf{u}\) \(\mathbf{u}+(-\mathbf{u})=-\mathbf{u}+\mathbf{u}=\mathbf{0},\;\text{where}\;-\mathbf{u}=(-1)\mathbf{u}\)

\(c(\mathbf{u}+\mathbf{v})=c\mathbf{u}+c\mathbf{v}\) \((c+d)\mathbf{u}=c\mathbf{u}+d\mathbf{u}\) \(c(d\mathbf{u})=(cd)\mathbf{u}\) \(1\mathbf{u}=\mathbf{u}\)

Example: Consider the vectors \(\mathbf{a}_1=\begin{pmatrix}1\\0\\-1\end{pmatrix}\), \(\mathbf{a}_2=\begin{pmatrix}1\\1\\0\end{pmatrix}\), \(\mathbf{b}=\begin{pmatrix}1\\2\\1\end{pmatrix}\) and \(\mathbf{c}=\begin{pmatrix}1\\2\\3\end{pmatrix}\). Are \(\mathbf{b}\) and \(\mathbf{c}\) linear combinations of \(\mathbf{a}_1\) and \(\mathbf{a}_2\)?

\[x_1\mathbf{a}_1+x_2\mathbf{a}_2=\mathbf{b}\quad\Longleftrightarrow\quad x_1\begin{pmatrix}1\\0\\-1\end{pmatrix}+x_2\begin{pmatrix}1\\1\\0\end{pmatrix}=\begin{pmatrix}1\\2\\1\end{pmatrix} \quad\Longleftrightarrow\quad\begin{pmatrix}x_1+x_2\\x_2\\-x_1\end{pmatrix}=\begin{pmatrix}1\\2\\1\end{pmatrix}.\]

Note that this corresponds to a system of linear equations:

\[\left\{\begin{array}{rrcr}x_1&+x_2&=&1\\&x_2&=&2\\-x_1&&=&1\end{array}\right.\quad\Longrightarrow\quad \left(\left.\begin{matrix}1&1\\0&1\\-1&0\end{matrix}\;\right|\;\begin{matrix}1\\2\\1\end{matrix}\right) \sim\left(\left.\begin{matrix}1&1\\0&1\\0&1\end{matrix}\;\right|\;\begin{matrix}1\\2\\2\end{matrix}\right) \sim\left(\left.\begin{matrix}1&0\\0&1\\0&0\end{matrix}\;\right|\;\begin{matrix}-1\\2\\0\end{matrix}\right).\]

Hence we have: \(x_1=-1\) and \(x_2\), which implies that \(\mathbf{b}=-\mathbf{a}_1+2\mathbf{a}_2\). Similarly we have:

\[y_1\mathbf{a}_1+y_2\mathbf{a}_2=\mathbf{c}\quad\Longleftrightarrow\quad\left\{\begin{array}{rrcr}y_1&+y_2&=&1\\&y_2&=&2\\-y_1&&=&3\end{array}\right. \quad\Longrightarrow\quad\left(\left.\begin{matrix}1&1\\0&1\\-1&0\end{matrix}\;\right|\;\begin{matrix}1\\2\\3\end{matrix}\right) \sim\left(\left.\begin{matrix}1&1\\0&1\\0&1\end{matrix}\;\right|\;\begin{matrix}1\\2\\4\end{matrix}\right) \sim\left(\left.\begin{matrix}1&0\\0&1\\0&0\end{matrix}\;\right|\;\begin{matrix}-1\\2\\2\end{matrix}\right).\]

We conclude that this system is inconsistent, which implies that \(\mathbf{c}\) is not a linear combination \(\mathbf{a}_1\) and \(\mathbf{a}_2\).

Since both computations above are more or less similar (the same row operations), we could have combined the two computations:

\[\left(\left.\begin{matrix}1&1\\0&1\\-1&0\end{matrix}\;\right|\;\begin{matrix}1&1\\2&2\\1&3\end{matrix}\right) \sim\left(\left.\begin{matrix}1&1\\0&1\\0&1\end{matrix}\;\right|\;\begin{matrix}1&1\\1&2\\2&4\end{matrix}\right) \sim\left(\left.\begin{matrix}1&0\\0&1\\0&0\end{matrix}\;\right|\;\begin{matrix}-1&-1\\2&2\\0&2\end{matrix}\right).\]

Note that we have

\[x_1\mathbf{a}_1+x_2\mathbf{a}_2+\cdots+x_n\mathbf{a}_n=\mathbf{b}\quad\Longrightarrow\quad\bigg(\mathbf{a}_1\;\mathbf{a}_2\;\ldots\;\mathbf{a}_n\;\bigg|\;\mathbf{b}\bigg).\]

Hence, an augmented matrix can be seen as the description of a system of linear equations (rows) and as the description of a vector equation (columns).

Definition: If \(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_p\) are vectors in \(\mathbb{R}^n\), then

\[\text{Span}\{\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_p\}:=\{c_1\mathbf{v}_1+c_2\mathbf{v}_2+\cdots+c_p\mathbf{v}_p\;|\; c_1,c_2,\ldots,c_p\in\mathbb{R}\}\]

denotes the set of all linear combinations of \(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_p\). This is the subset of \(\mathbb{R}^n\) spanned or generated by the vectors \(\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_p\).

In the example above we have seen that \(\mathbf{b}\in\text{Span}\{\mathbf{a}_1,\mathbf{a}_2\}\) and \(\mathbf{c}\notin\text{Span}\{\mathbf{a}_1,\mathbf{a}_2\}\).

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Last modified on March 22, 2021