- #1
iJake
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Homework Statement
In the follow cases find a maximal linearly independent subset of set ##A##:
(a) ##A = \{(1,0,1,0),(1,1,1,1),(0,1,0,1),(2,0,-1,)\} \in \mathbb{R}^4##
(b) ##A = \{x^2, x^2-x+1, 2x-2, 3\} \in \mathbb{k}[x]##
The Attempt at a Solution
The first part of the exercise is trivial, as it is easy to observe that the second vector is a linear combination of the first and third vectors. My question is one of mechanics. Should I be writing the elements of each vector as the rows or columns of a matrix?
In a previous exercise, I had to determine whether or not a vector space was spanned by a set of vectors. In that case, it was:
Determine if ## V = \mathbb{k}[x]_3## is spanned by ##A = \{1, 1+x^2, 1-x+x^2+x^3, 4-x+2x^2+x^3\}##
I reordered the elements and presented them as the columns of a matrix:
##
\begin{matrix}
0 & 0 & 1 & 1\\
0 & 1 & 1 & 2\\
0 & 0 & -1 & -1\\
1 & 1 & 1 & 4\\
\end{matrix}
##
I did row operations (switch R4 with R1, then with R3, add R3 to R4) and found a row of 0s, thus telling me the rank of the matrix is 3 and that A does not span V.
However, for my current problem, I wrote the matrix using the vector elements as rows, not columns. When I write them as columns I do not reach the same conclusion, which confuses me.
##
\begin{matrix}
1 & 0 & 1 & 0\\
1 & 1 & 1 & 1\\
0 & 1 & 0 & 1\\
2 & 0 & -1 & 0\\
\end{matrix}
##
Here I clearly observe that the second row is the sum of the first and third rows. However, if I write it where the vector elements are columns, I see something else:
##
\begin{matrix}
1 & 1 & 0 & 2\\
0 & 1 & 1 & 0\\
1 & 1 & 0 & -1\\
0 & 1 & 1 & 0\\
\end{matrix}
##
Can someone explain the difference to me? Thanks.
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