Question Regarding Vector Spaces and Basis I was doing some practice problems with vector spaces and I don't understand this question:

Find a basis for the span of the following polynomials in the vector spaces of all polynomials:
  $$\{x^2 − 1, 2x − 3, x^2 + 1, 4\}.$$

I made each vector a column vector and put them together, and this is what I got:
$$\begin{pmatrix}
−1 & −3 & 1 & 4 \\
0 & 2 & 0 & 0 \\
1 & 0 & 1 & 0
\end{pmatrix}.$$
I reduced it and the pivots were in columns 1, 2, and 3. The book says that means the basis is $\{x^2 − 1, 2x − 3, x^2 + 1\}$, but I don't understand the relationship between the columns with pivots and the basis itself. What is it?
 A: One of the key facts about row-reduction is that it does not affect linear dependencies between columns. That is, any particular grouping of columns is linearly independent before row reduction exactly if that grouping is linearly independent after row reduction. Similarly, if you can write some column as a linear combination of some other columns after row reduction, you could already do so before reducing (with the same coefficients, even). When you find the pivot columns, you are finding columns that are linearly independent (before or after reduction). Also, it is possible to write the columns that aren't pivot columns as linear combinations of those that are.
The above reasoning means that, for any list of vectors, you can find a basis for the span of that list by:


*

*writing the vectors as columns of a matrix,

*row reducing to find the pivot columns,

*and concluding that a basis (but not necessarily the only one) is given by the original vectors that went in those columns,


since the reduction shows that they are independent and we can write the other vectors as combinations of them.
Finally, in your case, you can draw the conclusion you did about your polynomials because the columns in your matrix are just representations of those polynomials as vectors of their coefficients.
