I studied "vectors are special representation of matrices" but here on mathstackexchange many high rputational users say "matrices are special represtation of vector"

My Opinion is based on Manga book By Shin Takashi : "Vectors are special representation of matrices" Is an Introductory statement of vectors.Which i found very much true in many concepts of vectors like addition subtraction dot products and cross products.


closed as primarily opinion-based by InsideOut, Namaste, levap, Shailesh, user91500 Dec 17 '16 at 6:26

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  • $\begingroup$ Do you have a question and not a mere opinion you want to air? $\endgroup$ – Namaste Dec 17 '16 at 0:49
  • $\begingroup$ Question must be ok now? $\endgroup$ – Kislay Tripathi Dec 17 '16 at 9:29

When people say 'vectors' what they mean is 'elements of a vector space'. Certain collections of matrices ($n$ by $n$ matrices over $\mathbb{R}$ for example) form vector spaces and hence the matrices in these spaces are 'vectors'.

The idea of such 'vectors' is much more general than that of matrices - the continuous functions (over a closed interval in $\mathbb{R}$, mapping to $\mathbb{R}$) is an infinite dimensional vector space. The function $f(x)=x^2$ is a vector in this space. How does this relate to a matrix however? In $\mathbb{R}^n$ we can write a vector as a $1$ by $n$ matrix, however in this continuous function setting we can not write $f(x)$ as some 'restricted matrix'.

My guess is that your reason for thinking matrices are 'more general' is because as I wrote above, in $\mathbb{R}^n$, one can think of a vector as a $1$ by $n$ matrix. The point is that this idea doesn't generalise to more abstract settings. We can't always relate a specific matrix to a vector.

  • $\begingroup$ f(x)=x^2 is not a linear transformation $\endgroup$ – Kislay Tripathi Dec 15 '16 at 11:13
  • 5
    $\begingroup$ Yes, it's not a linear transformation; however, it is a vector in the vector space of continuous functions over a bounded interval. $\endgroup$ – Zestylemonzi Dec 15 '16 at 11:24
  • $\begingroup$ So any function that is continuous on some [a,b] is a vector and element of vector space over R $\endgroup$ – Kislay Tripathi Dec 15 '16 at 13:01
  • $\begingroup$ Yes, any continuous function over $[a,b]$ can be considered as a vector in the vector space consisting of all such functions. Polynomials (over one variable) with coefficients in $\mathbb{R}$ can also be considered as a vector space. You can consider polynomials as vectors in this setting. $\endgroup$ – Zestylemonzi Dec 15 '16 at 13:20
  • $\begingroup$ Why not write a continuous function as a column vector but with continuum many entries :)? $\endgroup$ – Oiler Dec 15 '16 at 15:12

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