向量方程与张成

Introduction

Vector Equations

Spans

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

Solve the vector equation

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

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

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).

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

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

Let $v_1$ and $v_2$ be two vectors in $\mathbb{R}^2$. The equality $\operatorname{Span}\{v_1,v_2\} = \mathbb{R}^2$ holds under which condition?

The vectors $(1,2)$ and $(2,4)$ span a line through the origin. The point $(3,6)$ lies on that line as $c\cdot(1,2)$. Enter the scalar $c$.