- #1

karush

Gold Member

MHB

- 3,269

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$\tiny{115.C51}$

find a linearly independent set T so that $\langle T\rangle =\langle S\rangle$

$S=\left\{

\left[\begin{array}{r}2\\-1\\2\end{array}\right],

\left[\begin{array}{r}3\\0\\1\end{array}\right],

\left[\begin{array}{r}1\\1\\-1\end{array}\right],

\left[\begin{array}{r}5\\-1\\3\end{array}\right]

\right\}$

make matrix A and derive RREF(A) to find pivot columns

$A=\left[

\begin{array}{rrrr}

2 & 3 & 1 & 5 \\

-1 & 0 & 1 & -1 \\

2 & 1 & -1 & 3

\end{array} \right]

\quad

\text{RREF}(A)=\left[

\begin{array}{rrrr}

1 & 0 & -1 & 1 \\

0 & 1 & 1 & 1 \\

0 & 0 & 0 & 0

\end{array} \right]$

\The pivot columns are observed at $C_1$ and $C_2$ thus we have

$T=\left\{

\left[\begin{array}{r}2\\-1\\2\end{array}\right],

\left[\begin{array}{r}3\\0\\1\end{array}\right]

\right\}$

then $\langle T\rangle =\langle S\rangle$

ok I think this is correct but I just followed a similar example

not sure just why this would be T=S when it looks like a subset

also, not up on the all the standard notatons for these type of problems

anyway mahalo much for any help

https://dl.orangedox.com/5LxJq55fJyIgYJE0y0

find a linearly independent set T so that $\langle T\rangle =\langle S\rangle$

$S=\left\{

\left[\begin{array}{r}2\\-1\\2\end{array}\right],

\left[\begin{array}{r}3\\0\\1\end{array}\right],

\left[\begin{array}{r}1\\1\\-1\end{array}\right],

\left[\begin{array}{r}5\\-1\\3\end{array}\right]

\right\}$

make matrix A and derive RREF(A) to find pivot columns

$A=\left[

\begin{array}{rrrr}

2 & 3 & 1 & 5 \\

-1 & 0 & 1 & -1 \\

2 & 1 & -1 & 3

\end{array} \right]

\quad

\text{RREF}(A)=\left[

\begin{array}{rrrr}

1 & 0 & -1 & 1 \\

0 & 1 & 1 & 1 \\

0 & 0 & 0 & 0

\end{array} \right]$

\The pivot columns are observed at $C_1$ and $C_2$ thus we have

$T=\left\{

\left[\begin{array}{r}2\\-1\\2\end{array}\right],

\left[\begin{array}{r}3\\0\\1\end{array}\right]

\right\}$

then $\langle T\rangle =\langle S\rangle$

ok I think this is correct but I just followed a similar example

not sure just why this would be T=S when it looks like a subset

also, not up on the all the standard notatons for these type of problems

anyway mahalo much for any help

https://dl.orangedox.com/5LxJq55fJyIgYJE0y0

Last edited: