I became to study linear algebra a bit on my own. It is already more than twenty years since my graduation. I ended up to my question due to practical problem that I have already solved. Having said that, my practical problem lead me to trying understand a concept of a column space, row space, null space and left null space.
I have developed a intuition or concept of a column space. In my thinking matrix columns are vectors in given space. Further on, in my thinking a column space is a space spanned by matrix columns. All that makes sense to me. I just rely on interpretation of a matrix as collection of vectors. Also, somehow I have become to understand a null space as a special space that will lead to matrix equation of Ax=0. In my thinking, importance of a null space is that there can't be inverse for a such transformation.
Now, my difficulties arise in understanding a row space. It is useless to repeat definition of a row space as Col(A'). The definition doesn't really open this for me. I'd like to understand significance of a row space. In my thinking, columns of a matrix are vectors. This understanding doesn't really fit to a row or a row space. Rows are not vectors, hence a row space is not similar space as a vector space spanned by vectors. Yes, technically it is so or may be, as I have understood. But I am still missing intuition or understanding of concept of a row space. What is significance of a row space, how does it manifest itself?
And very similarly, what is importance of a left null space, how does it manifest itself?
You see, I am not a mathematician nor student of math. I am an engineer. And due to my personal way of learning, I am always trying to visualize or develop some kind of intuition before diving into the details and rigorous definitions.
Thanks in advance
- Poincaré look-alike -