Equivalent Conditions for $B$ to be a Basis

The following conditions are equivalent to a subset $B \subseteq V$ being a basis:

<aside> 1️⃣ $B$ is linearly independent and $span(B)=V$

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<aside> 2️⃣ $B$ is a maximal linearly independent set.

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Maximal linearly independent:

1. It is linearly independent.
2. Appending any vector makes it linearly dependent.

<aside> 3️⃣ $B$ is a minimal spanning set.

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Minimal Spanning set:

1. It is spanning.
2. It is no longer spanning if we delete any vector.

How do we find a Basis?

We can find a basis by any one of the methods described below:

1. Start with the $\phi$ and keep appending vectors which are not in the span of the set thus far obtained, until we obtain a spanning set.
2. Take a spanning set and keep deleting vectors which are linear combinations of other vectors, until the remaining vectors satisfy that they are not a linear combination of the other remaining ones.