Lecture 6: Span of a Subset
Example 1
\((4,3) \in \mathbf{R}^2\). This vector can be written as
So \((4,3)\) is a linear combination of \((1,0)\) and \((0,1)\).
Example 2
\(x^3 + 3x + \pi \in P_3\). in a linear combination of \(x^3, x^2, x, 1\) with \(a_1 = 1, a_2 = 0, a_3 = 3, a_4 = \pi\).
Span of of a Subset
Example 3
Let \(S = \{\bar{0}\}\), then \(Span(S) = \{\bar{0}\}.\)
Example 4
Let \(S = V\), then \(Span(S) = V.\)
Example 5
Let \(S=\{(1,0),(0,1)\}\). \(S \subset \mathbf{R}^2\). Then,
Example 6
Let \(S=\{(1,m)\}\). \(S \subset \mathbf{R}^2\). Then,
This is the line through the origin with slope \(m\).
Example 7
Let \(S=\{(1,0,0),(0,0,1)\}\). \(S \subset \mathbf{R}^3\). Then,
This is the \(xz\)-plane in \(\mathbf{R}^3\).
Proof: Let \(S \subset V\) where \(V\) is a vector space. By definition we know that \(Span(S)\)
We will show that \(Span(S)\) is a subspace of (V) by verifying the three Subspace conditions.
- We need to show that \(\bar{0} \in Span(S)\). \(S\) is nonempty so we can choose some vector \(u \in S\). Since all linear combinations of \(u\) are in \(Span(S)\), then \(0u \in Span(S)\). But \(0u = \bar{0}\). Therefore, \(\bar{0} \in Span(S)\) and we're done.
- We need to show that \(Span(S)\) is closed under addition. Suppose \(u, v \in Span(S)\). We need to show that \(u + v \in Span(S)\). We can write \(u\) and \(v\) as follows
$$ \begin{align*} u &= a_1u_1 + a_2u_2 + ... + a_ku_k \\ v &= b_1u_1 + b_2u_2 + ... + b_ku_k. \end{align*} $$Then the sum can be written as,$$ \begin{align*} u+v &= (a_1+b_1)u_1 + (a_2+b_2)u_2 + ... + (a_k+b_k)u_k. \end{align*} $$So \(u+v \in Span(S)\) as we wanted to show.
- We need to show that \(Span(S)\) is closed under scalar multiplication. Similar to the previous argument we can see that
$$ \begin{align*} cu &= (ca_1)u_1 + (ca_2)u_2 + ... + (ca_k)u_k. \end{align*} $$So \(cu \in Span(S)\) as required.
From (a), (b), (c), we can conclude that \(Span(S)\) is a subspace of \(V\).
Example 7
Is \(x^3 - 3x + 5\) in \(Span(\{x^3 + 2x^2 - x + 1, x^3 + 3x^2 - 1\})\)?
For \(x^3 - 3x + 5\) to be in the span, this means that \(x^3 - 3x + 5\) can be written as a linear combinations of what’s inside the set. In other words, there exists \(a_1, a_2 \in \mathbf{R}\) such that
These equations are the same when the coefficients are equal. So equivalently we can solve the following system of equations:
We can use Gaussian Elimination to solve this system by doing a forward pass followed by a backward pass.
We’ll put the matrix in Row Echelon Form. Starting with zeroing out the entries below the first leading entry in the first column.
Continuing with the next leading entry to get.
This tells us that there is a solution so the answer is yes, we can write it \(x^3 -3x + 5\) is in the span above.
Example 8
The set \(\{(1,0),(0,1)\}\) generates \(\mathbf{R}^2\).
Example 9
The following set
generates the vector space of 2x2 symmetric matrices. How do we go about proving this?
Consider the matrix:
If \(A\) was symmetric, then we must have \(a_{12} = a_{21}\) or \(a_{ij} = a_{ji}\) if \(i \neq j\). This means we can re-write \(A\) as
References:
- Math416 by Ely Kerman