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show that the vector (2,7,6) be we written as a linear combination of the vectors

(1,3,2) and (0,1,2)

(b) show that the vector (-1,0,4) can be written as a linear combination of the vectors (1,3,2) and (0,1,2)

(c)

show that Span((1,3,2),(0,1,2)) = Span( (2,7,6), (-1,0,4))

My solution (a).

I write vectors in equation form

[tex]x_1 \[ \[ \begin{array}{c} 1 \\ 3 \\ 2 \end{array} \] + x_2 \[ \begin{array}{dd} 0 \\ 1 \\ 2 \end{array} \] =

\[ \begin{array}{c} 2 \\ 7 \\ 6 \end{array} \] [/tex]

which can be rewritten to

[tex] \[ \begin{array}{ccc} x_{1} \\ 3x_{1} + x_{2} \\ 2x_{1} + 2x_{2} \end{array} \] = \[ \begin{array}{c} 2 \\ 7 \\ 6 \end{array} \][/tex]

There must exist x_1 and x_2 which makes the above set of equations true.

These are found be rewritten the system into its equivalent coefficient matrix.

[tex] \[ \begin{array}{ccc} 1 & 0 & 2 \\ 3 & 1 & 7 \\ 2 & 2 & 6 \end{array} \] [/tex]

using row reduction I get

[tex] \[ \begin{array}{ccc} 1 & 0 & 2 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{array} \] \mathrm[/tex]

which gives x_1 = 2 and x_2 = 1

If I insert into the original equation, then values of x_1 and x_2 make it true.

(b)

Following the same method used in (a) I get x1 = -1 and x_2 = 3.

(c)

How do I use these results to prove that Span((1,3,2),(0,1,2)) = Span((2,7,6),(-1,0,4)) ???

Can I claim the set of vectors are dependent, and therefore their spans equal each other?

Best Regards

Fred

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