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My apologies for asking ambiguous question.

[Edited]What i know about basis of a vector space/subspace($\mathbb{R}^n$) is the bunch of vectors $v_{1},v_{2},...,v_{1}$ such that

1) They are independent.

2) They span the space.

Why actually we are interested in finding the basis of a space? I am saying that independent column vectors of a matrix(provided the column vectors span that space) provide the basis of that space. What does that basis mean w.r.t to that matrix?

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    $\begingroup$ Can you give some more context? I suspect you don't mean a basis of a matrix but rather a basis of the image of the corresponding linear mapping? $\endgroup$ – Hirshy Aug 14 '15 at 8:09
  • $\begingroup$ In proving properties of a vector space, it usually suffices to prove this for a basis, which is a lot easier. $\endgroup$ – rwols Aug 15 '15 at 8:48
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I think an example will be great to clarify your doubt. So here is an example-

Let say, $\{X : X \in \mathbb{R}^{2\times2}\}$. So, one basis for this space is-

$$\Big\{\begin{bmatrix} 1 & 0 \\0 & 0\end{bmatrix},\begin{bmatrix} 0 & 1 \\0 & 0\end{bmatrix}, \begin{bmatrix} 0 & 0 \\1 & 0\end{bmatrix},\begin{bmatrix} 0 & 0 \\0 & 1\end{bmatrix}\Big\}.$$

Definition of basis of a vector subspace: The set of minimum number of vectors to span the vector subspace is called a basis for the vector space. Reference- Wikipedia.

Let assume a matrix, $$A =\begin{bmatrix} 1 & 0 \\0 & 0\end{bmatrix}.$$ The range space of this matrix is a subspace of $\mathbb{R}^2$. So the basis for the range space is only $\Big\{\begin{bmatrix} 1 \\0\end{bmatrix}\Big\}$ whereas a basis for $\mathbb{R}^2$ is $\Big\{\begin{bmatrix} 1 \\0\end{bmatrix},\begin{bmatrix} 0 \\1\end{bmatrix}\Big\}.$

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