Section 3 Definitions

Basis
Let H be a subspace of a vectorspace V
An indexed set of vectors $\beta = \text{span } {\vec{b}_1,\vec{b}_2,...,\vec{b}_p}$ each in V is a basis for H if

1. $\beta$ is a linearly independent set
2. H = span{$\vec{b}_1,\vec{b}_2,...,\vec{b}_p$}

For $\Re^3$, the most basic basis is the 3x3 identity matrix, $\beta =$ span{$\begin{bmatrix} 1\\0\\0 \end{bmatrix}, \begin{bmatrix} 0\\1\\0 \end{bmatrix}, \begin{bmatrix} 0\\0\\1 \end{bmatrix}$}

These vectors are linearly independent, and linear combinations of them can make any vector in $\Re^3$, therefore, it is a basis.

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