Linear Algebra

When given multiple linear equations and asked to solve for certain variables, the first requirement is that you have as many equations as you have variables. If you have more equations than variables, that's fine; just use whichever are the simplest. If you have more variables than equations, the best you can do is solve for one variable in terms of another.

In a scholastic setting, you would normally have the same number of variables as equations. That is what will be assumed for the remainder of this article. The most common format of a system of linear equations is this:

y=2x+3

y=x+5

When asked to solve these equations, what's really being asked is for you to find where those two lines intersect. This allows you to assume that the x value and y value will be the same at that point. Therefore, you can say that

y=2x+3=x+5

2x+3=x+5

x=2

Then, since we know that x=2, we can put that value into either of the initial equations to solve for y. Let's choose the second one, because then we don't have to worry about multiplying by two.

y=x+5

y=2+5

y=7

Therefore, the two lines intersect at the coordinates (2,7). This method is normally referred to as solving by comparison, which is a special case of solving by substitution (see below).

Sometimes, equations are not formatted in a way that solving by comparison is convenient. For example, perhaps something like this is given:

6=9y-3x

2y=3x+4

In situations like these, it is often necessary to do some algebra before combining the equations in some way. Let's try to isolate the variable x in the first equation.

6=9y-3x

3x+6=9y

3x=9y-6

x=3y-2

Now that we have x in terms of y, let's put that new equation into the second given equation.

2y=3(3y-2)+4

2y=9y-6+4

2y=9y-2

2=7y

y=2/7

Again, putting this y value into the first equation, we get:

6=9y-3x

6=9(2/7)-3x

6=18/7-3x

6-18/7=3x

42/7-18/7=3x (in this step, we multiplied 6 by 7/7)

24/7=3x

24/7*(1/3)=x

x=24/21

Therefore the coordinates that these two lines intersect at is (24/21,2/7).

You'll notice that this set of coordinates are less appealing than the integers found in the first problem. Since substitution is a very general method of solving linear equations, it will work under any circumstances. If you have doubts about using comparison (or the final method, outlined below) then use substitution. Provided you don't make any mistakes in the mechanics, substitution will get you the correct answer.

The final method is called elimination. Using this method, one can often avoid long and tedious algebra. For example:

2x+y=1

-2x+2y=5

If we were to use substitution, we would get the correct answer (though we'd have to isolate a variable, expand a bracket, isolate for the other variable in terms of the first, substitute that equation into the other, and eventually solve for both variables). However, there is an easier way. We can add the two equations together.

2x+(-2x)+y+2y=1+5

0+3y=6

y=2

Substituting y=2 into the first equation, we get:

2x+y=1

2x+2=1

2x=-1

x=-(1/2)

Therefore the coordinates that these two lines intersect at is (-1/2,2)

It is worth noting that we are not limited to adding the two equations together. We may also subtract, multiply, and divide the equations. Elimination often takes the most practice to spot when it will be useful, but it can save many lines of math when applied in a suitable manner (as in the last example).

Those are the basic methods for solving two equation, two unknown problems. They will also hold true for three or more equation/unknown problems, but the mechanics of the solution becomes very long, very quickly. In more than two equation/variable questions, it is recommended to use matrices, but that is beyond the scope of this answer.

The indefinite integral of (1/x^2)*dx is -1/x+C.

Yes, a vertical line is linear, but it is not a function, because every point on the line has the same x value.

Matrices were used to organize data.

7-3

If you graph a Linear equation it will be a strait line. If it doesn't come out strait, its not linear.

Also a linear equation can be put into y=mx+b, with mx meaning the slope and b meaning Y-intersept.

When you have a fraction on each side of the equals you can multiply the denominator of the left side times the numerator of the right. It will equal the product of the numerator on the left side times the denominator of the right.

That is called cross products.

You see the point the two lines cross, if they do. This is the solution to the system since it is the values of (x,y) that are on both lines The solution is a sytems is those points, if any, (x,y) that satisfy both equations. That is the same as saying they are on both lines. If you graph the equations, this is the same as saying the points that are in the intersection of the lines. This is why parallel lines represent a system with no solution and if two equations are the same line there is an infinite number of solutions.

A system of linear equations determines a line on the xy-plane. The solution to a linear set must satisfy all equations. The solution set is the intersection of x and y, and is either a line, a single point, or the empty set.

A coefficient is a number before a variable. For example, in 2x, the 2 would be the coefficient

rise over run.

A linear actuator is a device that creates linear motion, often by converting rotational motion. For instance, an electric linear motor turns a screw that in turn "pushes" a cylinder. By this action, linear motion is created from a rotating shaft.

An idempotent is a matrix whose square is itself. Specifically, A^{2}=A. For example the 2x2 matrix

A= 1 1

0 0

is idempotent.

The assertion is true.

Let A be an idempotent matrix. Then we have A.A=A. Since A is invertible, multiplying A-1 to both sides of the equality, we get A = I.

Q. E. D

The transpose of a matrix A is the matrix B that is obtained by swapping the rows and columns of A into the columns and rows of B.

In algebraic form, if A = {aij} then B = {aji} is its transpose, where 1 ≤ i ≤ n and 1 ≤ j ≤ m.

If a big triangle has a base length of 6, and a small triangle has a corresponding base length of 3, the scale factor from large to small is 2/1 (or 2). The scale factor from small to large would be 1/2.

Recall that if a matrix is singular, it's determinant is zero. Let our nxn matrix be called A and let k stand for the eigenvalue. To find eigenvalues we solve the equation det(A-kI)=0for k, where I is the nxn identity matrix.

(<==)

Assume that k=0 is an eigenvalue. Notice that if we plug zero into this equation for k, we just get det(A)=0. This means the matrix is singluar.

(==>)

Assume that det(A)=0. Then as stated above we need to find solutions of the equation det(A-kI)=0. Notice that k=0 is a solution since det(A-(0)I) = det(A) which we already know is zero. Thus zero is an eigenvalue.

Oh honey, no. Linear means a straight line relationship between two variables, like y = mx + b. But 2πr is the formula for the circumference of a circle, not a straight line. So, in short, 2πr is definitely not linear.

Linear sums mean in a line. so, for example;

3+2=5 is linear. 3+

2

_

5 is column addition. Hope this helps.

(I know it's clumsy it's hard on the computer!)

One of the most common ways to represent linear equations is to use constants. You can also represent linear equations by drawing a graph.

Likening the equals sign to a see-saw may help. Young people tend to have physical experience with teeter-totters and understand the idea that balancing requires equal weight on either end. For older young people, a set of scales can greatly help as a visual aid.

Why am I wasting my time with such an obvious question? Multiplication Property of Zero! If you multiply anything by zero, the product is zero!

This is known as the "Difference".