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I have a basic doubt. Can we say that a set of vectors span the entire vector space iff they are linearly independent ? Do they need to satisfy any other property ?

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    In addition to that you need enough vectors in your set! For example the set $\{(1,1,0)\}$ of vectors of $\mathbf{R}^3$ is linearly independent, but only generates a 1-dimensional subspace. If your space is finite dimensional, then it suffices to check that, in addition to linear independence, the number of vectors in your set equals the dimension of the space. In an infinite dimensional space you need to show that any vector is a linear combination of vectors in your linearly independent set.2011-06-21

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