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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?

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  • $\begingroup$ How do you represent a vector in 3-dimensional space? $\endgroup$
    – angryavian
    Apr 23, 2020 at 0:18
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    $\begingroup$ 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. $\endgroup$ Apr 23, 2020 at 0:22
  • $\begingroup$ 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. $\endgroup$ Apr 23, 2020 at 0:45
  • $\begingroup$ @CyclotomicField It's still a vector over the field of real numbers, any tuple of real numbers can be considered a vector. $\endgroup$
    – David Reed
    Apr 23, 2020 at 3:33
  • $\begingroup$ @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. $\endgroup$ Apr 23, 2020 at 11:49

2 Answers 2

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I believe you are used to think of vectors as pointed arrows. The length of the arrow gives the magnitude of the vector. The direction of the arrow gives the direction of the vector. The column/ row matrix representation of vectors can be arrived by thinking like this. If you have a pointed arrow in a plane (a piece of paper), draw a pair of perpendicular lines , namely the X and Y axis, such that the origin (point of intersection of X and Y axis ) should be the starting point of the pointed arrow vector you have on your paper.

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.

The abstract way of thinking of vectors is as pointed arrows.. Once coordinates are given to the space, these abstract vectors can be thought of as points in the space represented by row/column matrices.

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The abstract answer is: a vector is a member of a vector space. A vector space is any set that has two operations (addition and scalar multiplication) satisfying certain requirements (the vector space axioms). Your geometric vectors form one vector space. Arrangements of numbers in a row or column of a certain size (or, for that matter, in any other arrangement) form another. And there are many more. For example, some of the most useful applications of linear algebra come from the fact that solutions of systems of homogeneous linear differential equations form a vector space.

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