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I know the basis of a vector space/subspace ($\mathbb{R}^n$) is it's the bunch of vectors $v_{1}, v_{2}, ..., v_{n}$ such that:

  1. They are independent.

  2. They span the space.

Why are we actually 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?

mins
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Sanjeev
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    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? – Hirshy Aug 14 '15 at 08:09
  • In proving properties of a vector space, it usually suffices to prove this for a basis, which is a lot easier. – rwols Aug 15 '15 at 08:48

2 Answers2

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

Community
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Rajat
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I am saying that independent column vectors of a matrix (provided the column vectors span that space) provide the basis of that space

Indeed the column vectors of your matrix form a basis for the vector space, but this basis is not unique, many different matrices can be used. For example $\small \begin {bmatrix} -1 & 2 \\ -3 & 4 \end {bmatrix}$ and $\small \begin {bmatrix} 6 & -5 & 4 \\ -3 & 2 & 1 \end {bmatrix}$ have both column vectors spanning $\mathbb R^2$. Note the column vectors of these matrices are not orthogonal, nor have a unit length and they come in different quantities.

Why actually we are interested in finding the basis of a space?

When looking for the basis, we mean the standard basis, which vectors are in minimum quantity (the dimension of the space), orthogonal, and unit.

For $\mathbb R^2$ it is $\left \{ \begin {bmatrix} 1 \\ 0 \end {bmatrix} , \begin {bmatrix} 0 \\ 1 \end {bmatrix} \right \}$.

It is indeed possible to get the standard basis from each of the previous matrices, finding the RREF of this matrix and using the appropriate number of columns.

mins
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