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What does a subspace say A spanned by another subspace B and a vector x mean ? Does that imply anything about a basis or does it just mean that every vector in subspace A is either present in subspace B or can be expressed as linear combination of vectors from B and x. Or anything else perhaps ?

Any help would be much appreciated.

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up vote 6 down vote accepted

To say that $A$ is spanned by $B$ and $x$ means that every vector in $A$ can be written as a linear combination of some vectors belonging to $B$ and perhaps $x$. If you want, you could choose any basis $\{v_1, \ldots, v_k\}$ of $B$, and then every vector $u \in A$ can be written as $$u = \alpha_1v_1 + \dots + \alpha_kv_k + \beta x,$$ for scalars $\alpha_1, \ldots, \alpha_k, \beta$. Note, however, that $\{v_1, \ldots, v_k, x\}$ is not necessarily a basis for $A$. This happens if and only if $x$ doesn't belong to $B$. If $x$ is already in $B$, then $A = B$, as you can easily check.

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