Bases as Coordinate Systems

Introduction

Let $\mathcal{B} = \{v_1, v_2\}$ with $v_1 = (1,1)$ and $v_2 = (1,-1)$, a basis of $\mathbb{R}^2$. A vector $x$ has $[x]_\mathcal{B} = (4,1)$. Which of the following is $x$?

Let $v_1 = (1,0,1)$, $v_2 = (1,1,1)$ and $V = \operatorname{Span}\{v_1,v_2\}$, a plane in $\mathbb{R}^3$. Which vectors $x$ in $\mathbb{R}^3$ have $\mathcal{B}$-coordinates $[x]_\mathcal{B}$, that is, for which $x$ does the equation $x = c_1v_1 + c_2v_2$ have a solution?

A subspace $V$ has a basis $\mathcal{B} = \{v_1,v_2,v_3\}$. In how many ways can a vector $x$ in $V$ be written as a linear combination of $v_1,v_2,v_3$? Enter a number.

Let $v_1 = (1,0,1)$, $v_2 = (1,1,1)$ and $x = (5,3,5)$. Enter the $\mathcal{B}$-coordinates of $x$ as two numbers $c_1,c_2$ (for example 1,2).

The pivot columns of $A = \left(\begin{array}{cccc} 1 & 1 & 2 & 4 \\ 1 & 2 & 2 & 5 \\ 1 & 3 & 2 & 6 \end{array}\right)$ are $v_1 = (1,1,1)$ and $v_2 = (1,2,3)$. Enter the coefficients $c_1,c_2$ with $x = c_1v_1 + c_2v_2$ for the third column $x = (2,2,2)$ of $A$, as two numbers (for example 1,2).

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