We can use the Sage tools we already have, along with Theorem LIVRN, to determine if sets are linearly independent. There is just one hitch — Sage has a preference for placing vectors into matrices as rows, rather than as columns. When printing vectors on the screen, writing them across the screen makes good sense, and there are more mathematical reasons for this choice. But we have chosen to present introductory linear algebra with an emphasis on the columns of matrices — again, for good mathematical reasons. Fortunately, Sage allows us to build matrices from columns as well as rows.
Let us redo Example LDHS, the determination that a set of three vectors is linearly dependent. We will enter the vectors, construct a matrix with the vectors as columns via the column_matrix() constructor, and analyze. Here we go.
Notice that we never explicitly row-reduce A, though this computation must happen behind the scenes when we compute the list of pivot columns. We do not really care where the pivots are (the actual list), but rather we want to know how many there are, thus we ask about the length of the list with the function len(). Once we construct the matrix, the analysis is quick. With $n\neq r$, Theorem LIVRN tells us the set is linearly dependent.
Reprising Example LIS with Sage would be good practice at this point. Here is an empty compute cell to use.
While it is extremely important to understand the approach outlined above, Sage has a convenient tool for working with linear independence. .linear_dependence() is a method for vector spaces, which we feed in a list of vectors. The output is again a list of vectors, each one containing the scalars that yield a nontrivial relation of linear dependence on the input vectors. We will give this method a workout in the next section, but for now we are interested in the case of a linearly independent set. In this instance, the method will return nothing (an empty list, really). Not even the all-zero vector is produced, since it is not interesting and definitely is not surprising.
Again, we will not say anymore about the output of this method until the next section, and do not let its use replace a good conceptual understanding of this section. We will redo Example LDHS again, you try Example LIS again. If you are playing along, be sure v1, v2, v3 are defined from the code above.
The only comment to make here is that we need to create the vector space QQ^5 since .linear_dependence() is a method of vector spaces. Example LIS should proceed similarly, though being a linearly independent set, the comparison with the empty list should yield True.