Does least squares solution to Ax=b depend on choice of norm

[kaxzr]

To find the closest point to $b$ in the space spanned by the columns of $A$ we have:
$$\mathbb{\hat{x}}=(A^TA)^{-1}A^T\mathbb{b}$$
My question is, shouldn't this solution $\hat{x}$ depend on the choice of distance function over the vector space? Choosing two different distance functions might give two different $\hat{x}$s. But this equation does not make any reference to the choice of distance function.

Can anyone explain this to me? This is not directly a homework question but I am just trying to get a better understanding of the concepts here.

Thanks.

 

[mfb]

If nothing else is given, use the standard scalar product and its induced norm.

 

[Ray Vickson]

To find the closest point to $b$ in the space spanned by the columns of $A$ we have:
$$\mathbb{\hat{x}}=(A^TA)^{-1}A^T\mathbb{b}$$
My question is, shouldn't this solution $\hat{x}$ depend on the choice of distance function over the vector space? Choosing two different distance functions might give two different $\hat{x}$s. But this equation does not make any reference to the choice of distance function.

Can anyone explain this to me? This is not directly a homework question but I am just trying to get a better understanding of the concepts here.

Thanks.

This function is *assuming* standard Euclidean distance. Of course for other measures of distance you will get different results. Some other norms do not lead to explicit solutions, but are only doable numerically.