Linear algebraic equation
Definition
An algebraic equation in the unknowns x1, x2, … , xn ( n>= 1) is linear if it can be written in the form a1 x1 + a2 x2 + … + an xn = b Where the coefficients a1, a2, …, an and b are usually given/known constants. The unknowns are sometimes called variables.
Goal
To find or solve the values of the unknowns x1, x2, … xn that satisfy the equation.
Example
2x = 10 Linear in the sense that the unknowns in the equation are linear.
Systems
A system of linear equations is a set of linear equations. Given in problems since some can’t be solved with just one equation, like 2a +3o = 45. Even when there is only one equation, it is often called a system. For… reasons?
Verbal problems
Translate the statements into equations and put them in a set.
Solutions
A solution of a system of equations in one choice of the values of unknowns that make all of the equations true/satisfied simultaneously. A solution set of a system of equations is the collection of all possible solutions. Solving a system of equations means finding the solution set.
Process to find solutions involves gaussian eliminations. To do so, you can subtract one equation from the other. For example, in the set of x1 -2x2 = -3, x1 - x2 = 2, we do the second - first x2 = 5, x1 = 7
Set of 4x + 4y = -10 equiv to x + y = -5/2 -8x -8y = 20 equiv to x + y = -5/2 y = -x -5/2
From a geometric POV, a solution is a point of intersection of the two equations represented as lines on a graph.
A system of linear equations is consistent if it has at least one solution. It is inconsistent if it has no solutions.
Vector
Vectors are in opposition to scalars, which are effectively numbers, they would change the magnitude of a Vector but not the direction
By definition
Let R be the set of real numbers. Then R^n is the set of ordered n-tuples of real numbers:
R^n = {(x1, x2, …, xn) | xi is a real number for all i = 1, 2,… n}.
An element of R^n is a point in R
No clue what this actually means.
For two points A = (a1, … , an) and B = (b1, …, b2) in R^n the Vector AB denotes the displacement from A to B and is defined by AB = (b1 - a1, b2 - a2, … bn - an). You can do Vector addition geometrically by drawing a parallelogram and taking its diagonal or adding them sequentially.
Vectors are generally not defined by the point of origin of the Vector, rather they are relative from the location of the head to the tail.
Multiplying by a number will increase the magnitude of a Vector, not the direction. This is a scalar. A negative multiplication will reverse the direction of the Vector.
Alt Def
In physics and geometry, is often referred to as a quantity that has a magnitude and a direction, except for 0 Vector that does not have a direction. THis is contrary to a scalar, a real number.
Problem examples
u = (1 5) v = (-1 1) w = (5 1) solve x in 7u -v + x = 6x + w
Qs
Can vectors be non-linear? What does it mean for a problem to be open style? Are all points vectors? Are all vectors points?