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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
Why is the stiffness matrix symmetric?
(I'm assuming you're referring to FEM)
The entries of a stiffness matrix are inner products (bilinear forms) of some basis functions. Insofar as you will typically be dealing with symmetric bilinear forms, the stiffness matrix will also be symmetric. In other words, ai,j = <φi,φj> = <φj,φi> = ai,j.
The issue is closely related to so-called "Gramian matrices" which, in addition to symmetry, have other properties desirable in the context of FEM. I've provided links below.
Why are systems of linear equations so important?
Systems of linear equations are so important because they give you an easy way to do mass and complex mathematical calculations. For example:
x/2 + y/4 = 255,
3x - y/2 = 100.
The first step is to solve the first equation for y:
2x + y = 1020,
y = 1020 - 2x.
Substitute that value of y into the second equation and solve for x:
3x - (1020 - 2x)/2 = 100,
3x - 510 + x = 100,
4x = 610,
x = 152.5.
Finally, substitute that value of x into the first equation to get the solution for y:
152.5/2 + y/4 = 255,
y/4 = 178.75,
y = 715.
Lets say that the equation is: 2+3x+2x=22
- 2+5x=22 Combine Like Terms.
- 2-2+5x=22-2 Subtract 2 from each side.
- 5x=20 Simplify.
- 5x/5=20/5 Divide 5 on each side.
- x=4 Simplify.
Check...
- 2+3x+2x=22 x=4
- 2+3(4)+2(4)=22 Substitute x for 4.
- 2+12+8=22 Multiply and Divide first from left to right.
- 14+8=22 Add and Subtract from left to right.
- 22=22
Should you have an equation with higher order (like x2) then different methods, like factoring or the quadratic formula will apply.
Eh?
How do you solve simultaneous equations?
simultaneous equations
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: === === : === === === === # # # # # === ===
: === === : === === : === === : === === : === ===
Linear prediction is a mathematical operation on future values of an estimated discrete time signal. Its rule is to predict the output by using the given inputs.
How do you make equations equivalent?
You cannot change two existing, completed equations so that they are equal to each other. However, when working with two equations, you may set them equal to each other to solve a system of equations. An example is the system of 2x+5y=103x-5p=10 You may now combine the two, as they both are equal to ten. This results in the eqation of 2x+5y=3x-5p You may simplify this to 5y=x-5p This brings you one step closer to solving, and one may complete the system with some additional information.
What are the differences between an idempotent matrix and a generalized inverse of a matrix?
Idempotent Matrix:
An idempotent matrix, A, is the specific periodic matrix (see note) where k=1, thus having the property A2=A (we can also say A.A=A).
Inverse Matrix:
Given a square matrix, A, its inverse is B if AB=BA.
Note:
A periodic matrix, A, has the property Ak+1=A where k is a positive integer. If k is the least positive integer for which Ak+1=A, then A is said to be of period k.
The phrase "idempotent matrix" is an algebraic term. It is defined as a matrix that equals itself when multiplied by itself.
What is the definition of an idempotent matrix?
A square matrix A is idempotent if A^2 = A. It's really simple
How do you take dot product of two vectors?
Multiply the product of their magnitudes by the cosine of the angle between them.
A scale where the distance between 1 and 2 is the same as the distance between 2 and 3; and so on.
By way of contrast, in a logarithmic scale, the distance between 1 and 10 would be the same as the distance between 10 and 100, or 100 and 1000.
There are many other scales.
How is Cramer's rule used in life?
Cramer's Rule is used to solve systems of linear equations by converting those equations into one matrix equation. For example:
ax + by = e
cx + dy = f
becomes
|a bx|=|e|
|c dy| |f|
An example that I came up with just yesterday involved cell phone bills. It looked something like this:
Cindy and Suzi just received their cell phone bills. They have the same provider and can see how many minutes they used and text messages, but they are curious to how much each text message and a single minute costs. Cindy's bill tells her that she used
638 text messages and 230 minutes for a total of $42. Suzi's bill showed that she used 290 text messages and 540 minutes for a total of $29. How much does a text message cost and how much does each minute cost?
To solve this question, you would set up the information given above into two equations:
638x + 230y = 42
290x + 540y = 29
where x is the cost of one text message and y is the cost of one minute used.
There are a lot of other examples in real life that you can set up like the following equations.
What is the definition of transpose in regards to a matrix?
The Transpose of a Matrix
The matrix of order n x m obtained by interchanging the rows and columns of the m X n matrix, A, is called the transpose of A and is denoted by A' or AT.
What is the definition of an anti-symmetric matrix?
The Definition of an Anti-Symmetric Matrix:
If a square matrix, A, is equal to its negative transpose, -A', then A is an anti-symmetric matrix.
Notes:
1. All diagonal elements of A must be zero.
2. The cross elements of A must have the same magnitude, but opposite sign.
What is the definition of a Hermitian matrix?
Hermitian matrix defined:
If a square matrix, A, is equal to its conjugate transpose, A†, then A is a Hermitian matrix.
Notes:
1. The main diagonal elements of a Hermitian matrix must be real.
2. The cross elements of a Hermitian matrix are complex numbers having equal real part values, and equal-in-magnitude-but-opposite-in-sign imaginary parts.
What is the definition of a skew-Hermitian matrix?
Skew-Hermitian matrix defined:
If the conjugate transpose, A†, of a square matrix, A, is equal to its negative, -A, then A is a skew-Hermitian matrix.
Notes:
1. The main diagonal elements of a skew-Hermitian matrix must be purely imaginary, including zero.
2. The cross elements of a skew-Hermitian matrix are complex numbers having equal imaginary part values, and equal-in-magnitude-but-opposite-in-sign real parts.
What makes two lines perpendicular?
Perpendicular lines are any two lines that intersect at a 90 degree angle.
What are the disadvantages of matrices?
Matrices are very difficult to use, especially with Matlab. This stress and trauma dramatically reduce your life expectancy by up to 3% in most cases, 5% if your last name is Eigen.
What is the dot product of two rectangular components of a vector?
a vector is a line with direction and distance. there is no answer to your question. the dot is the angular relationship between two vectors.
"Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another." (from Wikipedia)
What are the different kinds of systems of linear equations?
Standard form: Ax + By = C, where A and B are non-zero constants.
Slope-intercept form: y = mx + b, where m is the slope, and b is the y-intercept.
Cross product is not difine in two space why?
When performing the cross product of two vectors (vector A and vector B), one of the properites of the resultant vector C is that it is perpendicular to both vectors A & B. In two dimensional space, this is not possible, because the resultant vector will be perpendicular to the plane that A & B reside in. Using the (i,j,k) unit vector notation, you could add a 0*k to each vector when doing the cross product, and the resultant vector will have zeros for the i & jcomponents, and only have k components.
Two vectors define a plane, and their cross product is always a vector along the normal to that plane, so the three vectors cannot lie in a 2D space which is a plane.
Field Axioms are assumed truths regarding a collection of items in a field.
Let a, b, c be elements of a field F. Then:
Commutativity:
a+b=b+a and a*b=b*a
Associativity:
(a+b)+c=a+(b+c) and (a*b)*c = a*(b*c)
Distributivity:
a*(b+c)=a*b+b*c
Existence of Neutral Elements:
There exists a zero element 0 and identify element i, such that,
a+0=a
a*i=a
Existence of Inverses:
There is an element -a such that,
a+(-a)=0
for each a unequal to the zero element, there exists an a' such that
a*a'=1
