- #1

amolv06

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## Homework Statement

Given

[tex]x_{1}= \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix}, x_{2}= \begin{pmatrix} 3 \\ -1 \\ 4 \end{pmatrix}, x_{3}= \begin{pmatrix} 2 \\ 6 \\ 4 \end{pmatrix}[/tex]

Find the dimension of [tex]Span(x_{1}, x_{2}, x_{3})[/tex]

## Homework Equations

If V is a vector space of dimension n > 0 then any set of n linearly independent vectors spans V.

If the x vectors are combined into a matrix and the matrix is nonsingular then the vectors are linearly independent.

[tex]dim R^{3} = 3[/tex]

## The Attempt at a Solution

[tex]det \begin{vmatrix} 2 & 3 & 2 \\ 1 & -1 & 6 \\ 3 & 4 & 4 \end{vmatrix} = -28.[/tex]

This shows that the three vectors are linearly independent. This should mean that any vector in [tex]R^{3}[/tex] should be able to be represented as a linear combination of the three given vectors. Therefore, shouldn't [tex]Span(x_{1}, x_{2}, x{3})[/tex] have a dimension of 3? The answers section in the back of my book says it has 2 dimensions. I can't quite figure out why.