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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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