Understand Vectors: Math Made Simple

[fend]

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!

 

[HallsofIvy]

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?

Without knowing what the problem is, what information you are given, and what you are trying to achieve, I don't see how anyone can tell you why you are doing something!

How does this relate to the original system?

What "original system"?

Thank you for reading!

 

[slider142]

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.

 

[fend]

When I originally posted this I could think of no better way to phrase this with my limited understanding of the subject matter. The question was about vectors in general and what exactly they are. Why in something like a simple 2x2 matrix would I plot the values of the columns to get a vector.

HallsofIvy: I appreciate the amount of questions you have answered and wealth of mathematical knowledge you possess, however you are a snob. I have posted two times on these forums and you have berated and criticized my posts. I do not know why you spend your time helping people when it is apparent that you have little respect for them in the first place. If you are the shining example of this forums help, than perhaps ignorance is bliss.

 

[fend]

Thank you very much slider142

 

[Mark44]

You can represent a system of linear equations by an equation that involves a matrix and a couple of vectors.

For example, this system:
2x + 3y + z = 5
x - y - 2z = 8
x - 4y + z = 12

can be represented as Ax = b, where A is the 3 x 3 matrix of coefficients of x, y, and z, and x is the vector consisting of variables, and b is the vector consisting of the constants on the right sides of the equations.

$$\left[\begin{array}{ccc} 2 & 3 & 1 \\ 1 & -1 & -2\\ 1 & -4 & 1 \end{array}\right]\left[\begin{array}{c } x \\ y \\ z \end{array}\right] =\left[\begin{array}{c } 5 \\ 8 \\ 12 \end{array}\right]$$

 

[trambolin]

The biggest problem of yours (which was also mine) is that you want to link the physical world to the abstraction that we use to solve the problem directly. And it will never(subjective opinion) happen. Read through the stories about determinant and its physical relevance or Laplace transform. You will never get a satisfactory (subjective opinion) story. The reason is that the abstraction uses some internal steps (e.g. infinite sums etc.) which is not necessarily happening in the physical world. Hence, try to distinguish the solution from the real world problem at hand. And ONLY look for the physical similarities or understandable steps AFTER you master the solution method.

I don't know how correct http://acharya.iitm.ac.in/mirrors/vv/vidya/emathist.html" is but even if it is not, the story solved many parts of the mystery about your original problem.

 

[fend]

Thank you Mark44 and trambolin for your responses. I feel much more secure with the idea of vectors now. Excellent job!

 

[Pi-Bond]

If you watch from 10:00, there is a pretty nice explanation of how vectors relate to their equations and the Cartesian plane.