Tazmin Horton

2020-11-02

Two different ways to express $\left[\begin{array}{c}1\\ 1\end{array}\right]$

as a linear combination of${v}_{1},{c}_{2}\text{}and\text{}{v}_{3}.$

as a linear combination of

Daphne Broadhurst

Skilled2020-11-03Added 109 answers

Given vectors are ${v}_{1}=\left[\begin{array}{c}1\\ -3\end{array}\right],{v}_{2}=\left[\begin{array}{c}2\\ -8\end{array}\right]\text{}and\text{}{v}_{3}=\left[\begin{array}{c}-3\\ 7\end{array}\right]$

Calculation:

Let the vector w =$$\left[\begin{array}{c}1\\ 1\end{array}\right]$$ .

The vector w can be represented in terms of as$w=a{v}_{1}+v{v}_{2}+c{v}_{3}.$

Determine the coefficients a, b, c.

Write the vectors${v}_{1},{v}_{2}\text{}and\text{}{v}_{3}$

as the columns of a matrix$[{v}_{1}{v}_{2}{v}_{3}\text{}2]$ and reduce the matrix in to row reduce echelon form.

$\left[\begin{array}{cccc}1& 2& -3& 1\\ -3& -8& 7& 1\end{array}\right]{R}_{2}\to 3{R}_{1}+{R}_{2}\left[\begin{array}{cccc}1& 2& -3& 1\\ 0& -2& -2& 4\end{array}\right]$

$\left[\begin{array}{cccc}1& 2& -3& 1\\ 0& -2& -2& 4\end{array}\right]{R}_{2}\to -\frac{1}{2}{R}_{2}\left[\begin{array}{cccc}1& 2& -3& 1\\ 0& 1& 1& -2\end{array}\right]$

$\left[\begin{array}{cccc}1& 2& -3& 1\\ 0& 1& 1& -2\end{array}\right]{R}_{1}\to {R}_{1}-2{R}_{2}\left[\begin{array}{cccc}1& 0& -5& 5\\ 0& 1& 1& 2\end{array}\right]$

This gives the following equations,

$a-5c=5$

$b+c=-2$

Here, c is a free variable.

Two find two different ways to write w as a linear combination of${v}_{1},{v}_{2},{v}_{3},$ choose two different values of c, which will give two different values of corresponding a and b.

Let,$c=1$ , this gives,

$a=10$

$b=-3$

Then, the vector w can be expressed as$w=10{v}_{1}-3{v}_{2}+{v}_{3}.$

Let$c=2,$ this gives,

$a=15$

$b=-4$

Then, the vector w can be expressed as$w=15{v}_{1}-4{v}_{2}+2{v}_{3}.$

Therefore, two different ways of expressing$\left[\begin{array}{c}1\\ 1\end{array}\right]$

as a linear combination of${v}_{1},{v}_{2},{v}_{3}\text{}are\text{}10{v}_{1}-3{v}_{2}+{v}_{3}\text{}and\text{}15{v}_{1}-4{v}_{2}+2{v}_{3}.$

Calculation:

Let the vector w =

The vector w can be represented in terms of as

Determine the coefficients a, b, c.

Write the vectors

as the columns of a matrix

This gives the following equations,

Here, c is a free variable.

Two find two different ways to write w as a linear combination of

Let,

Then, the vector w can be expressed as

Let

Then, the vector w can be expressed as

Therefore, two different ways of expressing

as a linear combination of

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