Linear Independence

Introduction

The Definition of Linear Independence

Criteria for Linear Independence

Pictures of Linear Independence

Linear Dependence and Free Variables

The vectors $v_1,\ldots,v_k$ are said to be linearly independent when which statement holds?

Which of the following sets of vectors is linearly dependent?

If some nontrivial linear combination of $v_1,\ldots,v_k$ equals the zero vector, the vectors are said to be linearly ______ .

The vectors $v_1 = \begin{pmatrix}1\\2\\1\end{pmatrix}$ and $v_2 = \begin{pmatrix}2\\4\\2\end{pmatrix}$ are dependent. Enter weights $x_1,x_2$, not both zero, such that $x_1v_1 + x_2v_2 = 0$ (for example: 2,-1).

Consider the six vectors in $\mathbb{R}^4$

Enter the largest possible number of independent vectors you can choose from this list (a number).

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