115.C51 find a linearly independent set T so that T=S

In summary, to find a linearly independent set T that generates the same subspace as S, we can use the row reduction method to find the pivot columns of the given matrix A. These pivot columns will form the basis for the subspace generated by S. In this case, we have pivot columns at $C_1$ and $C_2$, which correspond to the first two columns of A. Therefore, the vectors in S corresponding to these pivot columns, namely $\left[\begin{array}{r}2\\-1\\2\end{array}\right]$ and $\left[\begin{array}{r}3\\0\\1\end{array}\right]$, will form a linearly independent set T that
  • #1
karush
Gold Member
MHB
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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
 
Last edited:
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  • #2


Hello,

Thank you for sharing your work on this problem. Your solution is correct. To find a linearly independent set T that generates the same subspace as S, we can use the row reduction method to find the pivot columns of the given matrix A. These pivot columns will form the basis for the subspace generated by S. In this case, we have pivot columns at $C_1$ and $C_2$, which correspond to the first two columns of A. Therefore, the vectors in S corresponding to these pivot columns, namely $\left[\begin{array}{r}2\\-1\\2\end{array}\right]$ and $\left[\begin{array}{r}3\\0\\1\end{array}\right]$, will form a linearly independent set T that generates the same subspace as S.

As for your question about why T=S, it is because T is a subset of S. In other words, the vectors in T are also present in S, but T may not contain all the vectors in S. In this case, T only contains two out of four vectors in S. However, these two vectors are sufficient to generate the same subspace as S.

Also, the standard notation for this type of problem is to use the angle brackets $\langle \rangle$ to denote the subspace generated by a set of vectors. So $\langle T\rangle$ represents the subspace generated by the vectors in T.

I hope this helps clarify any confusion. Keep up the good work!
 

FAQ: 115.C51 find a linearly independent set T so that T=S

1. What does it mean to find a linearly independent set?

Finding a linearly independent set means identifying a group of vectors that are not dependent on each other, meaning that no vector in the set can be expressed as a linear combination of the other vectors in the set.

2. Why is it important to find a linearly independent set?

Linear independence is important because it allows us to simplify and solve complex systems of equations, and it also helps us to understand the relationships between different vectors in a system.

3. How do you determine if a set of vectors is linearly independent?

To determine if a set of vectors is linearly independent, you can use the Gaussian elimination method or the determinant method. If the determinant of the matrix formed by the vectors is non-zero, then the vectors are linearly independent.

4. What is the process for finding a linearly independent set?

The process for finding a linearly independent set involves using the Gaussian elimination method or the determinant method to identify the linearly dependent vectors in a set. Then, these dependent vectors can be removed to create a set of linearly independent vectors.

5. How is finding a linearly independent set useful in real-world applications?

Finding a linearly independent set is useful in many real-world applications, such as in engineering, physics, and computer science. It allows us to solve complex systems of equations and model real-world phenomena, such as electrical circuits and physical forces.

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