fend said:
I have spent a good amount of time trying to understand how vectors relate to a system of linear equations. Much of the information I find online are proofs which simply confuse me. I love math, but it is an extremely difficult relationship for me. If anyone could put vectors into simple terms, it would greatly help me begin deeper exploration of them.Questions that I am blurry on are...Why am I plotting points constructed from columns of my matrices? How does this relate to the original system? .
Thank you for reading!
A vector, at your level, is a common way to quantize orientation or another type of quality in a multidimensional space, or any proprietary sequence of values that may vary.
Ie., an RGB vector is usually used in image manipulation programs to specify the color of a pixel using 3 intensity values of the primary colors red, green, and blue.
CMY is another "coordinate system" to the same space, using different primary colors (basis vectors).
Since each color can be quantized, we can also embed color space in a 3-dimensional space.
Systems of equations can be about many things. In a manner similar to the above, we know from geometry that linear equations in n variables are (n-1)-dimensional planes in n-dimensional space. The coefficients of the variables are the components of a normal vector to the (n-1)-plane with respect to a basis consisting of unit vectors, one for each variable.
The points on a plane are by definition the solutions to the corresponding equation. If you have two equations, and thereby two planes, the points lying on the intersection of the two planes satisfy both equations. It is not a far reach to see that there may be geometric methods using vectors that aid in the algebraic solution of the equations, and vice versa.
There is a direct correspondence between the geometry and the algebra.
Matrices can represent two things at your level. At some point, you may prove their equivalence.
The first that you seem to have encountered is creating a matrix from the coefficients of a system of linear equations. From our previous discussion, you should be able to see that the rows of this matrix are the normal vectors of each plane in the graph of the equations. From this, you can see many things.
For example, if we have 3 2-dimensional planes in 3-dimensional space (the graph of a system of 3 equations with 3 unknowns), there are a few ways where we know that no unique solution exists (where the planes do not all intersect at a single point).
Ie., two planes may be parallel (collinear normal vectors). Or the normal vectors may all lie in the same plane (coplanar normal vectors). There is an algebraic method of determining whether these two things occur using algebra of the vectors (which are the rows of your matrix), which you will learn in your course.
