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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
Is this equation linear or nonlinear x2 plus y3 4?
Without an equality sign the given terms can't be considered to be any kind of an equation.
How can check whether a solution to any linear equation is correct?
to find a linear equation the roots must have been given in the question. to check whether its correct or not use this method. x2-(SOR)x+POR=0. SOR = sum of roots, POR = product of roots. if your SOR and POR is similar to your final answer, then the solution is correct
Why is the distributive property needed to solve algebraic equations?
because it tells you that you need to do distribute that number to all the other numbers. if you didnt use it your would come out with the wrong answer... not all the time but in most cases your answer will be incorrect if you do not use the distributive property
A linear equation is when each term in the algebraic equation is either a constant or the product has a single variable and a constant.
What is the answer for 7x-4y equals 217?
The answer comprises infinitely many points on a straight line.
When can you say that the equation is linear?
An equation is linear when it contains only variables of degree 1 and constants.
ALL linear equations will be of the form:
a1x1+a2x2+a3x3+...+anxn=c where an and c are constants.
How do you add alerbriac expressions to each other?
Adding two equations is an easy step. First choose a variable you want to eliminate. Multiply each one by the number that would get you the opposite or additive inverse so it will be gone from the modified equation.
x+y=5 =>3x+3x=15
2x-3x=-5 =>2x-3x =-5
5x =10
x = 2
Substitute that number into one of the equations and solve for the other.
2+y=5
y=3
What is a system of linear equations that has no solution?
there is no linear equations that has no solution every problem has a solution
How do you find linear equations with just coordinates?
if we take the (x1,y1),(x2,y2) as coordinates the formula was (x-x1)/(x2-x1)=(y-y1)/(y2-y1)
The first is 2-dimensional, the second is 1-dimensional.
What are applications for matrices?
Equations are an algebraic way of writing down a maths problem in shorthand.
Two or more simultaneous equations may be used to describe the same problem.
Matrices can be used to solve these simultaneous linear equations (that is equations with two or more unknown variables) and obtain the answer to those unknowns which satisfies both.
Equations are therefore generally solved to get values of unknown variables.......
Variable values are calculated (or assumed) to know all working or constant parameters of a system...
e.g. for a chemical reaction; generally pressure, temperature, concentration of reactant etc., may be combinations of unknown variables.
i.e. If these parameters are varied resultant yield get affected.........
We never know all properties at start, we first found relations between variables by doing practicals & form equations.........
Then these equations can be solved by many methods.......
Out of these many methods matrices is one......
So which ever system can be represented by equations, matrices have application there........
e.g. engineering problems, weather forecasting, aerospace design, financial calculations, chemical processes, construction calculations etc...........
And.....they were used by Albert Einstein to come up with his theories for General and Special Relativity.
Steps in solving problems involving systems of linear equations?
The answer depends on the level of your knowledge. The High level, simple answer is first. The Low level slog follows:
HIGH LEVEL, SIMPLE
Suppose you have n equations of the form
a11x1 + a12x2 + ... + a1nxn = bn where
the as are coefficients,
x1, x2, ... xn are the unknown variables
and
b1, b2, ... bn are the constants.
Write the n linear equations in n unknowns in the form Ax= b
where
A is an n*n matrix of coefficients
x is the n*1 matrix of the unknown variables
and
b is the n*1 matrix of the constants.
Find the inverse of A.
Then x = A-1b.
The above method works if the system has a unique solution. If the n equations are not independent, you will need to use a generalised inverse and that starts to get rather complicated. If they are inconsistent, then neither the inverse nor generalised inverse will be found.
LOW LEVEL SLOG
Use the first equation to express x1 in terms of the other variables. Substitute this value for x1 in the remaining n-1 equations. You now have n-1 equations in n-1 unknown variables.
Use the first of the new equations to express x2 in terms of the other variables. Substitute in remaining equations. You now have n-2 equations in n-2 unknown variables.
Continue until you have 1 equation in 1 unknown.
That will be of the form pxn = q so that xn = q/p.
Substitute this value into one of the equations at the 2-equations-in-2-unknowns stage. That will give you xn-1.
Work your way back to the top.
The two methods are equivalent. There are shortcuts available for matrix inversion (eg using determinants), but these are too complicated to go into here.
What are the different kinds of linear equation?
There can be linear equations with 1, 2, ... variables. Each of these is different since an equation with n variables belongs to n-dimensional space.
Can you create a system of linear equations from your own life?
A landscaping company placed two orders with a nursery. The first order was for13bushes and4trees, and totalled$487. The second order was for6bushes and2trees, and totalled$232. The bills do not list the per-item price. What were the costs of one bush and of one tree?
Letb =the number of bushes andt = the number of trees and set up a system of equations:
first order:13b+ 4t= 487
second order:6b+ 2t= 232
Example equations of linear equations?
y=3x+2
y-4x=9
These are examples of linear equations which is a first degree algebraic expression with one, two or more variables equated to a constant.
So x=2 is a linear equation as is y=1 but x2 =1 is not since the variable, x , has degree 2.
How many solutions can a system of linear equations with two variables?
None, one or infinitely many.
It is a certain type of equation used in Algebra. It uses letters to replace numbers and the aim of cracking the equation is to find the value of the letter. Such as:
4b+2=26 ....you would need to find the value of 'b'.
-2( 4b= 24 )-2
( b=6 )divided by 4
I've just solved the equation.
What is the equation of an algebraeic linear equation?
Y=MX+B B is the y-intercept. M is the slope Slope= y1+y2 _______ x1+x2 Or Rise over Run. X is the amount you multiply by usally. The amount of time. Ex. 3/1x+12=y x=5 So, y=28.
What piecewise linear transformation functions are used in image processing?
The form of the piecewise functions can be arbitrarily complex, but higher degrees of specification require considerably more user input.
