I have just started reading Linear Algebra and there are some basic things I cannot understand.
I read some answers on this site and also tried to search in some books but I didn't find a clear answer.
Here are few words form my textbook :
1. Here by VECTOR we do not mean the vector quantity which we have defined in vector algebra as a directed line segment.
2. Matrices having a single row or column are referred to as vectors.
3. I also watched a video in which at approximately 3:55 he says that a point in two dimensional real coordinate space is written in matrix form in LINEAR ALGEBRA.
Now I'm confused! Since in the video it seems as a vector in LINEAR ALGEBRA is just same as a point in 1, 2, 3…n real coordinate spaces. But in my text book its written that vector is not the vector quantity!
- And I also read in a book in which it was written that we are using LINEAR word instead of VECTOR to avoid confusion!
So is it really the difference in words? Why its not written clearly what the vector really is.
Best Answer
A vector by definition is an element of some vector space. Unless specified otherwise, this is the definition you should have in mind. Now let me try to clear up some of your specific questions.
Without context, it's impossible for me to figure out exactly what is being meant here. Most likely, they previously introduced a specific vector space, such as $\mathbf{R}^n$ and now they want to discuss a different vector space where direction may not have a clear definition.
This is a bit more advanced than what you are probably studying. It basically comes down to how matrices actually arise. Once you fix a basis for your vector space, there is a bijective correspondence between linear transformations and matrices. Then all matrices arise as such. The proof involves taking a basis for the domain and then the columns (or rows) are the images under this map. Well, the image is an element of the codomain, i.e. an element of a vector space, so we can call it a vector.
This way, we can see that all columns of such a matrix is a vector of the codomain. For rows, now just switch the codomain and the domain.
This is going back to 1, where we are once again working in what appears to be $\mathbf{R}^2$. He says it is more common to write the vector $(5,0) \in \mathbf{R}^2$ as a column matrix instead of as a point notation. This is merely a naming or a left vs. right ($xA = b$ vs. $Ax = b$ if you will) and has nothing to do with whether it's a vector.