A problem in my book asks us to show that two vector sets S1 = {(1, 2, -1), (0, 1, 1), (2, 5, -1)} and S2 = {(-2, -6, 0), (1, 1, -2)} have the same span. My book solved this by putting the vectors into matrices as follows:

\begin{bmatrix}1&2&-1\\0&1&1\\2&5&-1\end{bmatrix} and \begin{bmatrix}-2&-6&0\\1&1&-2\end{bmatrix}

They then reduced the matrices to reduced row echelon form, and drew the conclusion that the sets have the same span because the reduced matrices are the same.

I'm just a little confused on why this means they have the same span -- what is reducing the matrices to row echelon form doing, exactly? What does the row-echelon form of a matrix of vectors represent? Previously my book has worked with coefficient/augmented matrices to find a solution of a system of equations, not with matrices of vectors.

I'm also confused on why they put the vectors from the set into a matrix -- is that valid thing you can do? My book previously stated that you can represent a single vector as a 1 x n matrix -- does the same idea apply here with multiple vectors? They've never actually used this practice previously in the book or in any examples, so I'm a little confused as to how it's being used in this problem.

  • $\begingroup$ First part of this problem. Can you verify that the three given vectors in $S_1$ are linear dependent or independent? Because I see some trouble in the exercise here... $\endgroup$ – imranfat Dec 7 '16 at 3:00
  • $\begingroup$ They have to be dependent if S1 and S2 span the same subspace, but I don't think my book uses that in their solution? $\endgroup$ – dagny Dec 7 '16 at 3:04
  • $\begingroup$ Are the 3 vectors in $S_1$ lin dependent? $\endgroup$ – imranfat Dec 7 '16 at 3:04
  • $\begingroup$ They have to be, right? $\endgroup$ – dagny Dec 7 '16 at 3:06
  • 2
    $\begingroup$ Elementary row operations don't change the span of the set of rows (you should think about why this is – it's important to understand it). Therefore, you are looking at the same span after you get to reduced row echelon form as before you started reducing (since you can carry out the reduction using elementary row operations), so comparing the two reduced forms is the same as comparing the two original spans. $\endgroup$ – Gerry Myerson Dec 7 '16 at 3:33

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