In a given vector space \(V\), the set of all the linear combinations that are possible with the vectors \(v_1, v_1, …, v_n\) present in \(V\) is called the generating set or span of that vector space. You can consider the general meaning of span to understand this concept. In layman’s terms, a span of a vector indicates how wide a particular set of vectors can stretch. If \(S\) is the generating set of vector space \(V\), it can be denoted as \(V = span[S]\). The concept becomes more comprehensible with some examples.
Example where span of a vector is a line
Consider the vector \((1, 1)\). Think of all the linear combinations possible with this vector. As a reminder, vector \(x\) is linear combination of \(v\) if \(x = \lambda v, \lambda \in \mathbb{R}\). So, think of as many possible linear combinations as you can.
The figure above demonstrates possible linear combinations in red and the vector \((1, 1)\) in blue. This concludes that the span of the vector \((1, 1)\) is a line.
Example where span of a vector is a plane
Let us consider vectors in two dimensions: \((1, 0), (0, 1)\). All the vectors that exist in vectors in \(\mathbb{R^2}\) can be expressed as a linear combination of the given set of vectors. Hence, the vector set spans across the entire two-dimensional plane.
Basis of a vector
If you’ve heard of span of a vector, then the term ‘basis’ should ring a bell. You might have read that every vector space consists of basis and it is a set of independent vectors in a vector space. So the question arises, what is the difference between basis and span?
Fundamentally, both, basis and span, are subspaces of a vector space. However, the term ‘linear combination’ is what differentiates basis from span. In this article on linear independence, we saw that vectors are linearly dependent if they can be expressed as a linear combination of each other. [inlinetweet]Basis is a set of independent vectors. It is the minimal generating set that defines a vector space. Span is a set of linearly dependent vectors as it is a result of the linear combination of the basis. Every vector \(v_1 \in V \) is a linear combination of vectors in basis \(B\).[/inlinetweet]
In the example above where span is a plane, notice that the set of given vectors \((1, 0), (0, 1)\) are independent vectors. Hence they form the basis of \(\mathbb{R^2}\).
Determining basis for a given vector space
It is time to call the messiah of linear algebra: Gaussian elimination. We saw in earlier posts that Gaussian elimination lets us find out solution to a system of linear equations and determining if vectors are dependent. However, the mathematical operation has got more functionality. Basis of a vector can also be easily known using Gaussian elimination. When we convert a matrix to its row-echelon form, the pivot columns indicate the independent vectors for the set. Hence, these vector form the basis of the given vector space.
Consider the matrix with vectors \(v_1, v_2 and v_3\):
\(
\left[\begin{array}{ccc}
1 & 2 & 3 \\
2 & 5 & 8 \\
3 & 7 & 4
\end{array}\right]
\)
The question is to find the basis for this set of vectors. Reducing the matrix to its tow echelon form, we get
\(
\left[\begin{array}{ccc}
1 & 2 & 3 \\
0 & 1 & 2 \\
0 & 0 & -7
\end{array}\right]
\)
Since all the columns are pivot columns, the set of vectors is independent. Therefore, \({v_1, v_2, v_3}\) is basis of the given vector space.
Conclusion
In this article, we saw that linear combination of given set of vectors results in generating set of that vector space, and is the span of it. If some vector space is spanned by a set of vectors that are linearly independent, that set is the basis of the defined vector space. To find the basis of some given vector space, reduce the matrix into row-echelon form. The pivot columns form the basis of that space.