Which are bases?

  • Thread starter Ted123
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Homework Statement



Which of the following sets [itex]S[/itex] are bases for the vector space [itex]V=\mathbb{R}^3[/itex]?

(a) [itex]S=\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \right\}[/itex]

(b) [itex]S=\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \right\}[/itex]

(c) [itex]S=\left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} , \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix} \right\}[/itex]

(d) [itex]S=\left\{ \begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \\ -1 \end{bmatrix} , \begin{bmatrix} -1 \\ 1 \\ 1 \end{bmatrix} \right\}[/itex]

The Attempt at a Solution



By my reckoning, the only set that isn't a basis is (b) as it isn't linearly independent and [itex]\text{Span}(S)\neq V[/itex]. Correct?
 

Answers and Replies

  • #2
LCKurtz
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Since there are only 3 and the dimension of R3 is 3, you are correct that whether or not they are linearly independent determines whether they are a spanning set (basis).
 

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