MATH 240

Wed. September 4th, 2019

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1.3: Vectors

• Column vector -- a matrix with one column, e.g. $\begin{pmatrix} 3\\5\\7 \end{pmatrix}$
• Can be thought of as a point in space
• Represents a single solution, e.g. $\begin{pmatrix} 3\\5\\7 \end{pmatrix} \rightarrow \begin{cases} x_1=3\\x_2=5\\x_3=7 \end{cases}$

Adding vectors

• Matrices can be added together:
  • Matrices must have the same dimensions
  • Add corresponding elements together
    $\begin{pmatrix} 2&3\\5&7 \end{pmatrix} + \begin{pmatrix} 5&3\\7&8 \end{pmatrix} = \begin{pmatrix} 7&6\\12&15 \end{pmatrix}$
• Vectors are a subset of matrices so they can be added too:
  $\begin{pmatrix} 2\\3 \end{pmatrix} + \begin{pmatrix} 3\\7 \end{pmatrix} = \begin{pmatrix} 5\\10 \end{pmatrix}$

Scaling vectors

• Can multiply every element of a vector by a constant:
  $5\begin{pmatrix} 3\\5 \end{pmatrix} = \begin{pmatrix} 15\\25 \end{pmatrix}$
• Negative scalars will also flip the direction of the vector

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Span

$\text{Span}\{v_1, v_2\}=\{c_1v_1+c_2v_2 \text{ ($c_1, c_2$ are scalars)}\}$

• The span of a set of vectors is the set of all possible linear combinations of those vectors.
• Size the of span is infinite unless $v_1=v_2...=v_n=0$
• If one vector is a linear combination of other vectors in the set, then removing it from the set won't change the span at all.

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Properties of $\ \R^n$

For all $u,v,w$ in $\R^n$, and $c,d$ in $\R$:
$\begin{aligned} u+v&=v+u\\ (u+v)+w&=u+(v+w)\\ u+0=0+u&=u\\ u+(-u)&=0\\ c(u+v)&=cu+cv\\ (c+d)u&=cu+du\\ (cd)u&=c(du)\\ 1u&=u \end{aligned}$

$ux_1+vx_2=w\text{ has a solution iff } w \text{ is in Span}\{u,v\}.$

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Example.

Is $\begin{pmatrix} 1\\3 \end{pmatrix}$ in the span of $\begin{pmatrix} 1\\1 \end{pmatrix}$ and $\begin{pmatrix} 2\\2 \end{pmatrix}$?
$\begin{gathered} \begin{pmatrix} 1\\1 \end{pmatrix}x_1+\begin{pmatrix} 2\\2 \end{pmatrix}x_2=\begin{pmatrix} 1\\3 \end{pmatrix}\\ \begin{pmatrix} 1&2&1\\1&2&3 \end{pmatrix} \rightarrow \begin{pmatrix} 1&2&1\\0&0&2 \end{pmatrix} \end{gathered}$
The system has no solution; therefore $\begin{pmatrix} 1\\3 \end{pmatrix}$ is not in the span.