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Vector Equations and Spans

1. Introduction

Introduction

2. Vector Equations

Vector Equations

3. Spans

Spans

4. What a span looks like

Let $v_1$ and $v_2$ be two noncollinear vectors in $\mathbb{R}^3$ . Which of the following is $\operatorname{Span}\{v_1,v_2\}$ ?

5. Solving a vector equation

Solve the vector equation

x\begin{pmatrix}1\\2\end{pmatrix} + y\begin{pmatrix}2\\1\end{pmatrix} = \begin{pmatrix}5\\4\end{pmatrix}.

Which pair $(x,y)$ satisfies it?

6. Vocabulary: span

In one word: the collection of all linear combinations of a list of vectors is called the ______ of those vectors.

7. Is $b$ in the span?

Decide whether $b=\begin{pmatrix}9\\12\end{pmatrix}$ lies in $\operatorname{Span}\{v,w\}$ where $v=\begin{pmatrix}1\\2\end{pmatrix}$ and $w=\begin{pmatrix}4\\5\end{pmatrix}$ , by solving $xv + yw = b$ . Enter $x,y$ (for example: 1,2).

8. Parameterizing a plane

The plane $x - 3y - z = 12$ passes through $(12,0,0)$ . Every point of the plane can be written as

\begin{pmatrix}x\\y\\z\end{pmatrix} = \begin{pmatrix}12\\0\\0\end{pmatrix} + y\begin{pmatrix}\ ?\ \\1\\0\end{pmatrix} + z\begin{pmatrix}\ ?\ \\0\\1\end{pmatrix}.

Solve the equation for $x$ and enter the first component of the $y$ -direction vector (a number).

9. In your own words

In your own words, state the single most important definition or theorem of this section, and explain what it says.

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