Basis and Dimension

Introduction

Basis of a Subspace

Computing a Basis for a Subspace

The Basis Theorem

Which of the following sets is a basis of $\mathbb{R}^2$?

The matrix $A = \left(\begin{array}{cccc} 1 & 2 & 0 & -1 \\ -2 & -3 & 4 & 5 \\ 2 & 4 & 0 & -2 \end{array}\right)$ reduces to $R = \left(\begin{array}{cccc} 1 & 0 & -8 & -7 \\ 0 & 1 & 4 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right)$. Which set is a basis of $\operatorname{Col}(A)$?

Complete the sentence: every basis of a nonzero subspace $V$ contains the same number of vectors, and that number is called the ______ of $V$. Enter one word.

Enter the dimension of $V = \operatorname{Span}\{(1,2,3),\,(2,4,6)\}$. Enter a number.

The plane $x-2y+3z=0$ is the null space of the one-row matrix $\left(\begin{array}{ccc} 1 & -2 & 3 \end{array}\right)$, whose free variables are $y$ and $z$. Its two special solutions, with $(y,z)=(1,0)$ and $(y,z)=(0,1)$, form a basis of the plane. Enter the second special solution, the one with $y=0$ and $z=1$, as three numbers $x,y,z$ (for example 1,2,3).

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