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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).

A set of $7$ vectors in $\mathbb{R}^4$ is given. What can you conclude about its linear dependence?

Find numbers $c_1, c_2$ with $c_1 = 2$ such that

Enter the answer as $c_1,c_2$.