Answer:
b is the span of these three vectors:
[tex]\vec{b}=\begin{pmatrix}7\\ 3\\ 8\end{pmatrix}\\\vec{a}_{1}=\begin{pmatrix}1\\ 2\\ -4\end{pmatrix},\vec{a}_{2}=\begin{pmatrix}8\\ 5\\ 6\end{pmatrix},\vec{a}_{3}=\begin{pmatrix}3\\ 3\\ 0\end{pmatrix}[/tex]
Step-by-step explanation:
1) The point here is can this vector be written as a Linear Combination of the three other ones?
Writing it as matrices:
[tex]\vec{b}=\begin{pmatrix}7\\ 3\\ 8\end{pmatrix}\\\vec{a}_{1}=\begin{pmatrix}1\\ 2\\ -4\end{pmatrix},\vec{a}_{2}=\begin{pmatrix}8\\ 5\\ 6\end{pmatrix},\vec{a}_{3}=\begin{pmatrix}3\\ 3\\ 0\end{pmatrix}[/tex]
Since we want to know if that can be a written as linear combination then let's rewrite them. This time, as an expression of vectors.
[tex]\left\{\begin{matrix}x_{1}\vec{a}_{1}+x_{2}\vec{a}_{2}+x_{3}\vec{a}_{3}=\vec{b}\\ \end{matrix}\right.[/tex]
2) Or in other words can this equation be true?
[tex]\begin{pmatrix}1 &8 &3 \\ 2 & 5 &3 \\ -4&6 &0 \end{pmatrix}=\begin{pmatrix}7\\ 3\\ 8\end{pmatrix}[/tex]
3) Augmenting that, and solving it by Gaussian Elimination:
[tex]\left.\begin{pmatrix}1 &8 &3&|7 \\2 & 5& 3&|3\\ -4& 6 &0&|8 \\ \end{pmatrix}[/tex]
[tex]2R_{1}-R_{2}\\-4R_{1}+R_{3}[/tex]
[tex]R_{2}:-11\\8R_{2}\\R_{2}-R_{1}[/tex]
[tex]38R_{2}-R_{3}[/tex]
(...) Then proceed the rest of the gaussian Eliminations, until you get the pivots on left side of the Matrix.
[tex]\begin{pmatrix}1 &0 &0 &0 \\ 0 & 1 &0 & \frac{4}{3}\\ 0& 0 &1 & \frac{-11}{9}\end{pmatrix}[/tex]
[tex]S=\left ( x_1=0,x_2= \frac{4}{3},x_3=\frac{-11}{9}\right )[/tex]
The solution are the scalar variables that may or may not be the span of those three other vectors as in this equation:
[tex]\left\{\begin{matrix}x_{1}\vec{a}_{1}+x_{2}\vec{a}_{2}+x_{3}\vec{a}_{3}=\vec{b}\\ \end{matrix}\right.[/tex]
4) Finally, let's try it:
[tex]0\begin{pmatrix}1\\ 2\\ -4\end{pmatrix}+\frac{4}{3}\begin{pmatrix}8\\ 5\\ 6\end{pmatrix}+\frac{-11}{9}\begin{pmatrix}3\\ 3\\ 0\end{pmatrix}=\begin{pmatrix}7\\ 3\\ 8\end{pmatrix}\\\\\begin{pmatrix}7\\ 3\\ 8\end{pmatrix}=\begin{pmatrix}7\\ 3\\ 8\end{pmatrix}[/tex]
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