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Let V be a vector space over real numbers and A be a linearly independent subset of V . Prove that A is a basis for V if and only if it is a maximal linearly independent subset of V . (If A is a linearly independent subset of V, we say that it is a MAXIMAL linearly independent set if the addition of any vector at all to A will result in a set which is not linearly independent).

Martin Sleziak
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Martina
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    possible duplicate: http://math.stackexchange.com/questions/1070828/basis-of-a-vector-space-is-a-maximal-linearly-independent-set?rq=1 – Emilio Novati Oct 26 '16 at 10:20

2 Answers2

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First, let us suppose $A$ is a basis of $V$. Write $A = \{ v_1,...,v_n \}$. If we add any $w \neq v_i $ for all $i$ to $A$, we must show that $\{ v_1,...,v_n,w \}$ is linearly dependent. Since $A$ is a basis of $V$, then we can write

$$ w = \sum \alpha_k v_k $$

for $\alpha_k \in \mathbb{R}$. Now, notice

$$ \mathbf{0} = \sum \alpha_k v_k - w $$

which means not all coefficients of this equation are zero, thus $\{v_1,...,v_n,w\}$ is linearly dependent.

Now, you try the other direction.

ILoveMath
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Look here for one implication. For the other implication, note that any any vector from a vector space can be combined as a linear combination of the basis vectors, and try to conclude that adding any vector to the set of the basis vectors makes it linearly dependent.

Community
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cdwe
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