Question on Vectors and Linear equation

Question

I am more of a physics student than mathematics and I also thought that vectors represented only things with a direction. For example in the equation a+b=0 . I found everywhere that the solution is a vector, this didn't make sense to me as a and b don't have any specific direction but the solution is a ordered pair. I was doing a course in linear algebra and found out that vectors can represent things like colors also,this is taken from the book Introduction to vectors by Stephen Boyd. A 3-vector can represent a color, with its entries giving the Red, Green, and Blue (RGB) intensity values (often between 0 and 1). The vector (0, 0, 0) represents black, the vector (0, 1, 0) represents a bright pure green color, and the vector (1, 0.5, 0.5) represents a shade of pink. Now I also thought that vectors had a direction but this did not make sense in the parts of the linear equation in which we plot a graph of a vs b, we do not plot the points in 2D space, Can someone clear up the part of the linear equation of plotting the points in 2D space and the colors with no direction being vectors?

Answer 1

While students in introductory physics courses are often taught that a vector is an object with magnitude and direction, this is not actually the definition of a vector. The actual definition of vector is an element in a vector space, which is a set paired with a field that satisfies certain properties regarding addition and scalar multiplication. In fact, it doesn't actually need to have concepts of magnitude or direction. The reason physics classes often discuss vectors in terms of magnitude and direction is because, by "vector," they really mean an element in a Hilbert space, which is a special type of vector space that also has a norm (which is used to define magnitude) and an inner product (which is used to define angles between vectors, and by extension, direction).