Span & Span solutions

Vector Equation

Given vectors $\{\vec{a}_1,\vec{a}_2,\dots,\vec{a}_n,\vec{b}\}$ in $\mathbb{R}^m$, determine if $\vec{b}$ is a linear combination of $\vec{a}_1,\dots,\vec{a}_n$

This is equivalent to solving the vector equation:

Which is the same as checking if the augmented matrix

has a solution (via EROS).

Review

• $\vec{b}$ is a linear combination of $\vec{a}_1,\dots,\vec{a}_n$
• The vector equation $x_1\vec{a}_1+\cdots+x_n\vec{a}_n=\vec{b}$ has a solution.
• We answer this by reducing the augmented matrix to (reduced) echelon form.

Example

Suppose

We want $x_1\vec{a}_1+x_2\vec{a}_2+x_3\vec{a}_3+x_4\vec{a}_4=\vec{b}$

Augmented matrix:

Row-reduce (EROS) to:

The solution is a one-dimensional general solution

From Vectors to Span

• Linear combinations involve adding vectors.
• This is how we visualize addition of vectors:

We often have to consider all possible linear combinations of a given collection of vectors.

In solving systems with infinitely many solutions, the solutions involve

for any choices of $s,t$

Span

If $k=1$ so there’s only one vector, then $\text{Span}\{\vec{v}\}$ is just all vectors that are multiples of $\vec{v}$

i.e., $\{c\vec{v}\mid c\in\mathbb{R}\}$ (a line through $\vec{0}$ and $\vec{v}$).

Example The collection of vectors

for all possible choices of $s, t$ form the span

Visualization

Picture this span when $k=1,2$:

• $k=1$: span is a line through $\vec{0}$ and $\vec{v}$ → $\text{Span}\{\vec{v}\}=\{c\vec{v}\}$
• $k=2$: span is $\text{Span}\{\vec{u},\vec{v}\}=\{c_1\vec{u}+c_2\vec{v}\}$ (Figure 2)

Spanning Solutions

Back to example above:

General solution was

We can rewrite this as

So we can think of the set of all solutions as

where $\vec{p}=\begin{bmatrix}3/2\\0\\2\\1/2\end{bmatrix}$ is a point, and $\vec{d}=\begin{bmatrix}-1\\1\\0\\0\end{bmatrix}$ is the direction.

We can picture the solution as a line in the direction of the second vector, going through the point given by the first vector (In $\mathbb{R}^4$)