# How is a column or row matrix considered a vector?

I’m pretty new to linear algebra so please bear with me. I’m just looking for some clarification on what a vector is. So I’ve always known a vector to be a magnitude in some direction. Basically a line with some value in some direction. I’m now learning about row and column “vectors”, except I thought these would be matrices. Just for example, I have a vector of test grades, ranging from 0-100. There’s no linearity, just a bunch of different values in a column matrix. What makes this a vector, or why is this called a column vector. There’s multiple magnitudes and no direction. I’m confused on this terminology, can somebody help clarify?

• How do you represent a vector in 3-dimensional space? Apr 23, 2020 at 0:18
• To be a vector space it has to satisfy certain axioms. Test grades don't do that because, for example, there is no $-x$ such that $-x+x=0$ in such a space. Apr 23, 2020 at 0:22
• It's important to recognize the difference between a vector, and its components relative to a coordinate system. The graphical view of a line with an arrow is the vector. When you define a coordinate system, you can now describe the arrow/vector by its components relative to those axes, as a list of numbers. The vector is independent of the choice of coordinate system. The components of the vector are not. Apr 23, 2020 at 0:45
• @CyclotomicField It's still a vector over the field of real numbers, any tuple of real numbers can be considered a vector. Apr 23, 2020 at 3:33
• @DavidReed No, they're called tuples not vectors. Without vector axioms including closure under vector addition, subtraction and scalar multiplication it's just a tuple. There are no vectors without vector space operations. Apr 23, 2020 at 11:49

Now i hope you see that, we have a vector (pointed arrow) in a coordinate space. In this coordinate space, all vectors start from the origin and end at some point in the space. The original pointed arrow also starts from origin and ends at some point say $$(a,b)$$. There you have your column matrix representing the original pointed arrow vector. Every point in the coordinate space represents a vector with starting at point at origin and head of the arrow at the specific point. As you must be aware, points in a coordinate space can be represented by $$(x,y)$$ which can be seen as a column or row matrix.