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Generating set of a vector space.

Here too, we consider a vector space V and a set I (this will be a set of indices, generally of the form $I= \{ 1,2, \dots , n \}$, but not necessarily).

Definition 5.3.1   A family $ \{ \overrightarrow{u_i} ; \; i \in I \}$ of vectors in V is a generating set for V if every vector in V is a linear combination of the $\overrightarrow{u_i} $'s, i.e.

\begin{displaymath}\forall \overrightarrow{v}\in V, \exists (\alpha_i, i \in I) ...
...w{v} = \underset{i \in I}{\sum}\alpha_i \overrightarrow{u_i} .

Example 5.3.2   We prove that the vectors $\overrightarrow{u_1} = \begin{pmatrix}1 \\ 2 \end{pmatrix}$ and $\overrightarrow{u_2} = \begin{pmatrix}2 \\ 3 \end{pmatrix}$ form a generating set for $\mathbb{R} ^2 $.

Let $\overrightarrow{v} = \begin{pmatrix}x \\ y \end{pmatrix} \in \mathbb{R} ^2$. Then:

\begin{displaymath}\overrightarrow{v} =\alpha_1 \overrightarrow{u_1} + \alpha_1 ...
...ow \begin{cases}\alpha_1 = 2y-3x \\ \alpha_2 = 2x-y \end{cases}\end{displaymath}

Thus, any vector $\overrightarrow{v} $ in $\mathbb{R} ^2 $ is a linear combination of $\overrightarrow{u_1} $ and $\overrightarrow{u_2} $.

Proposition 5.3.3   Let F be a generating set for the vector space V. If G is a family of vectors such that $F \subseteq G$, then G is a generating set for V.

Hint: Add zeros as coefficients for the extra vectors.

Noah Dana-Picard