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Linear Algebra
Linear algebra is the detailed study of vector spaces. With applications in such disparate fields as sociology, economics, computer programming, chemistry, and physics, including its essential role in mathematically describing quantum mechanics and the theory of relativity, linear algebra has become one of the most essential mathematical disciplines for the modern world. Please direct all questions regarding matrices, determinants, eigenvalues, eigenvectors, and linear transformations into this category.
2,176 Questions
What is the relationship between linear algebra and economics?
Many problems in economics can be modelled by a system of linear equations: equalities r inequalities. Such systems are best solved using matrix algebra.
What are the three types of system of linear equations?
The three types are
- the system has a unique solution
- the system has no solutions
- the system has infinitely many solutions.
If A is any mxn such that AB and BA are both defined show that B is an mxn matrix?
By rule of matrix multiplication the number of rows in the first matrix must equal the number of rows in the second matrix. If A is an axb matrix and B is a cxd matrix, then a = d. Then if BA is defined, then c = b. This means that B is not necessarily mxn, but must be nxm.
How do you show a matrix is invertible?
For small matrices the simplest way is to show that its determinant is not zero.
Is linear algebra and linear equations the same?
Linear Algebra is a branch of mathematics that enables you to solve many linear equations at the same time. For example, if you had 15 lines (linear equations) and wanted to know if there was a point where they all intersected, you would use Linear Algebra to solve that question. Linear Algebra uses matrices to solve these large systems of equations.
What is the point at which the lines intersect in a system of linear equations?
The coordinates of the point of intersection represents the solution to the linear equations.
When is the product of the slopes of two perpendicular lines not equal to -1?
When the perpendicular lines are horizontal and vertical.
What is an independent system of linear equations?
An independent system of linear equations is a set of vectors in Rm, where any other vector in Rm can be written as a linear combination of all of the vectors in the set. The vector equation and the matrix equation can only have the trivial solution (x=0).
How do you define system linear equations?
A linear equation in n variables, x1, x2, ..., xn is an equation of the form
a1x1 + a2x2 + ... + anxn = y where the ai are constants.
A system of linear equations is a set of m linear equations in n unknown variables. There need not be any relationship between m and n. The system may have none, one or many solutions.
If you add two linear equations will you always get a linear equation?
One linear equation: Ax + By = C (A, B, and C are constants) Another linear equation: Dx + Ey = F (D, E, and F are constants) Their sum: (A+D)x + (B+E)y = (C + F) The coefficients (A+D), (B+E), and (C+F) are still constants, so the sum is still a linear equation.
Define the system of linear equation?
its an equation that you can graph and when the points are connected, it makes a line. usually includes variables x and y.
How do I solve for y the linear equations 8x-7y equals 7 and 7x equals 8y equals 8?
8x - 7y = 7 (A)7x + 8y = 8 (B)
7*(A): 56x - 49y = 49
8*(B): 56x + 64y = 64
8*(B)-7*(A): (64+49)*y = 64-49
=> 113y = 15
=> y = 15/113
What are the methods in solving systems of linear equation?
Three main methods come to mind:
- Substitution method
- Elimination method
- Matrix row-reduction.
The first two methods are taught in early grades in most high schools. Matrix row-reduction is taught in senior years in high school, or more likely, in the early years of a university math or science program.
In the substitution method, a variable is isolated (solved for) and then used to substitute in other equation(s).
e.g.
2x + y - 6 = 0
x + 3y - 13 = 0
Solve for y in the first equation. Move the 2x and -6 to the right side, switch signs as we switch sides.
y = -2x + 6
Now take this definition of y and substitute it in the second equation; any place you see y, make the substitution.
x + 3(-2x + 6) - 13 = 0
x + -6x + 18 - 13 = 0
Collect like terms and simplify.
-5x + 5 = 0
-5x = -5
x = 1
Substitute this value of x into the definition for y above.
y = -2(1) + 6
y = -2 + 6
y = 4
So the two lines given intersect at (1, 4).
In the elimination method, multiples of each equation are added (or subtracted) together, eliminating one variable at a time, when they add to 0. Let's use the same equations as before, but use the elimination method this time.
e.g.
2x + y - 6 = 0
x + 3y - 13 = 0
Multiply each term of the first equation by -3. You could just have easily multiplied the second equation by -2. Some of these numbers may look familiar.
-6x - 3y + 18 = 0
x + 3y - 13 = 0
Add the corresponding values in the equations.
-6x + x - 3y + 3y + 18 - 13 = 0 + 0
Adding the positive 3y and the negative 3y yields 0, thus eliminating the y value from further calculations.
-5x + 5 = 0
-5x = -5
x = 1
Now take that value and put it into one of the original equations and solve for the remaining variable.
2(1) + y - 6 = 0
2 + y - 6 = 0
y = -2 + 6 = 4
If we have done things properly, both methods should yield the same result. Which method is easier will depend on the exact specifics of the question.
These methods can be expanded for more than 2 unknown variables defined by 2 equations, but the amount of work required grows quite quickly. In those situations it is often more efficient to use a matrix and perform row reduction. Matrices are a complex topic, and so I won't cover them here. In many simpler situations, elimination and substitution will do the job quite nicely.
* * * * *
There is also the graphical method. Plot the linear equations and the coordinates of point of intersection of the lines is the solution.
What is the difference between positive and negative angles?
Essentially, none. Every negative angle can be made positive by adding 2*pi radians (or 360 degrees, or a multiple).
What is inconsistent system of linear equation?
It is a system of linear equations which does not have a solution.
How do you graph rise over run?
This is plotted with a straight line. The "rise" is how far the line rises vertically. the "run" is how far it traverses horizontally. The division "rise" / "run" is the "slope" of the line.
What are the steps in solving problems involving a system of linear equation?
For example, take these 2 equations: 2x+5y=25(eq 1) and 3x+4y=17(eq 2) First, since none of the variables have the same coefficient,you try to make them have it. How? Easy. You multiply both equations by a number till you get it. To solve, you multiply the whole first equation by 3 and the whole second equation by 2 which yields: 6x+15y=75 and 6x+8y=34. Then, you subtract the first equation from the second equation which yields: 7y=41 so y=41/7. Now you substitute in any equation:2x+5(41/7)=25 After simplification, x=-15/7. So there you go, x=-15/7 and y=41/7.
Explain why a system of linear equations cannot have exactly two solutions?
if you can fart out of your chin then you know your headin in the right direction
Why do square matrices only have multiplicative inverses?
there are pseudo inverses for non-square matrices
a square matrix has an inverse only if the original matrix has full rank which implies that no vector is annihilated by the matrix as a multiplicative operator
What is the definition of solution of system of linear equations?
The values for which the equations are solved.
Graphically the intersection of the lines that are the solutions to the individual equations. The link below gives some explanations. The equations themselves will have to be given for a solution to be found.
A = coefficient matrix (n x n)
B = constant matrix (n x 1)
