It is possible that one coordinated is not proportional to another, but is very nearly so. Some multiple m will bring a very close value. The vector mx is as close to y as possible The residual vector r is the distance btwn the tips of those two vectors. The better value of m, the smaller r will be because if mx was to equal y, r would be zero. By taking the dot product, r can be determined.
Data points often show a linear tendency put do not often lie on one single line, or are products of one another. All points can be expressed in three vectors where u contains the x coords, v the y-intercepts, and s the y coords. The best fit line is the linear combo of u and v that gives s where u and v are multiplied by multiple m, plus r, to make up for the distance btwn that line and s. The fundamental problem of linear modeling is to find a linear combo of vectors that are as close to vector b as possible, with each vector multiplied by a multiple x, and all added to perpendicular vector r, which would ideally be zero. Then, b is in the span of the u vectors.
Conflicts.
Because a line can be drawn through any two points, doesn't that make the two points proportional to each other?
Reflection.
Because the two vectors meet at the origin, the further they get the farther they are from each other. Therefore this concept seems very similar to local linear approximations.
This concept seems very relevant because almost always when using data, the points don't fall on one line, making a best fit line necessary.
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