Dottie Parra

2021-01-23

Give a correct answer for given question

(A) Argue why $\left(1,0,3\right),\left(2,3,1\right),\left(0,0,1\right)$ is a coordinate system ( bases ) for ${R}^{3}$ ?

(B) Find the coordinates of $\left(7,6,16\right)$ relative to the set in part (A)

Given $B=\left\{\left[\begin{array}{c}1\\ 0\\ 3\end{array}\right],\left[\begin{array}{c}2\\ 3\\ 1\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\right\}$

By definition, we say that tha vector $\left\{{v}_{1},{v}_{2},...,{v}_{n}\right\}$ are linearly dependent if there exists scalars ${\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{n}$ not all of them zero such that ${\alpha }_{1}{v}_{1}+{\alpha }_{2}{v}_{2}+\dots +{\alpha }_{n}{v}_{n}=\stackrel{―}{0}.$

We say that the vectors $\left\{{v}_{1},{v}_{2},...,{v}_{n}\right\}$ are linearly independent if ${\alpha }_{1}{v}_{1}+{\alpha }_{2}{v}_{2}+\dots +{\alpha }_{n}{v}_{n}=\stackrel{―}{0}$ then

The set of vectors $\left\{{v}_{1},{v}_{2},...,{v}_{n}\right\}$ is said to be a basis for vector space V if i) set of vectors $\left\{{v}_{1},{v}_{2},...,{v}_{n}\right\}$ is linearly independent ii) span $\left\{{v}_{1},{v}_{2},...,{v}_{n}\right\}=V$ If B = $\left\{{v}_{1},{v}_{2},...,{v}_{n}\right\}$ is a basis in a vector space V than every vector $\stackrel{\to }{v}\in V$ can be uniquely expressed as a linear combination of basis vectors ${b}_{1},{b}_{2},\dots ,{b}_{n}.$ i.e there exists unique scalsrs ${\alpha }_{1},{\alpha }_{2},\dots ,{\alpha }_{n}$ such that, $\stackrel{\to }{v}={\alpha }_{1}{b}_{1}+,{\alpha }_{2}{b}_{2}+\dots +{\alpha }_{n}{b}_{n}.$ The coordinates of the vector $\stackrel{\to }{v}$ relative to the basis $B$ is the sequence of co-ordinates, i.e. $\left[v{\right]}_{B}=\left({\alpha }_{1},{\alpha }_{2},..,{\alpha }_{n}\right)$

Consider $A=\left[\begin{array}{ccc}1& 2& 0\\ 0& 3& 0\\ 3& 1& 1\end{array}\right]$ Applying ${R}_{3}\to {R}_{3}-3{R}_{1},$ we get

Applying ${R}_{3}\to 5{R}_{2}+3{R}_{3},$ we get $\sim \left[\begin{array}{ccc}1& 2& 0\\ 0& 3& 0\\ 0& 0& 3\end{array}\right]$ (because in eacelon form, first, second and third columns have pivot elements)

$\therefore$ The set of vectors $\left\{\left[\begin{array}{c}1\\ 0\\ 3\end{array}\right],\left[\begin{array}{c}2\\ 3\\ 1\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]\right\}$ is linearly independent dim

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