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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
How do i solve systems of equations by adding or subtracting?
You multiply one equation by some constant, or both equations by different constants, and then add one equation to another (or subtract). Here is an example:
2x + 4y = 30
x + y = 10
Multiplying the second equation by (-2), you get:
2x + 4y = 30
-2x -2y = -20
Now you can add the two equations together, resulting in:
2y = 10
You can solve this for "y"; then you can replace the value you found in any of the original equations to find the corresponding value for "x".
Can linear equations and linear inequalities be solved the same way?
Yes, but with the following two rules to remember. If you multiply or divide both sides by a negative number, then the inequality sign is reversed (> becomes <, or < becomes >). Adding and subtracting numbers have no effect on the direction of the inequality. Also, if you have a 'not equals' sign, then it is unaffected by the multiplication.
The same is true if you take the reciprocal of both sides.
Example: with the equation: 1/x = 2, take the reciprocal and x = 1/2. With the inequality 1/x < 2, this becomes x > 1/2. You could also solve it by multiplying both sides by x, then dividing both sides by 2, and get 2 < x, which is the same as x > 2.
Another example: 3 - x > 7. Subtract 3 from both sides: -x > 4. Multiply both sides by -1: x < -4.
You could also go about this as: add x to both sides: 3 > 7 + x, then subtract 7 from both sides: -4 > x, which means the same as x < -4
The Mastodon Matrix is a research project that is being run by Cornell University. A mastodon was excavated in New York state and volunteers are required to sift through the earth surrounding the mastodon and to record their observations.
Follow link for more details.
What statement describes the function c equals 5g plus 8?
It is an equation in two variables, c and g.
What is gauss eliminating method?
Consider a system of linear equations . Let be its coefficient matrix. elementary row operation.
(i) R(i, j): Interchange of the ith and jth row.
(ii) R(ci): Multiplying the ith row by a non-zero scalar c.
(iii) R(i, cj): Adding c times the jth row to the ith row.
It is clear that performing elementary row operations on the matrix (or on the equations themselves) does not affect the solutions. Two matrices and are said to be row equivalent if and only if one of them can be obtained from the other by performing a sequence of elementary row operations. A matrix is said to be in row echelon form the following conditions are satisfied:
(i) The number of first consecutive zerosincreases down the rows.
(ii) The first non-zero element in each row is 1.
The process of performing a sequence of elementary row operations on a system of equations so that the coefficient matrix reduces to row echelon form is called Gauss elimination. When a system of linear equations is transformed using elementary row operations so the coefficient matrix is in row echelon form, the solution is easily obtained by back substitution.
How are metals used in everyday life?
for zinc u can use pennies,for silver u can use any kind of silverware,for aluminum u can use popcans,and for copper there are many cooking utensils made of it.
Can a pair of linear equation have exactly two solutions?
No.
A pair of linear equation can have 0 solutions (they are parallel), or one solution (they cross at one point) or an infinite number of solutions (they represent the same line).
What are the ways of finding the linear functions?
A linear function can be determined simply from its form. A linear function, over a set of variables, x1, x2, x3, etc is of the form
a1x1 + a2x2 + a3x3 + ... = b
where a1, a2, a3, etc and b are constants.
In graphical terms the linear function over n variables is a straight line in n-dimensional space.
In 2-d different letter are used instead of subscripts and the equation becomes ax + by = c
This can easily be shown to be equivalent to y = mx + c (though with a different value for c).
What is the set of values for which an equation or inequality is true?
It is called the solution set.
How do you find the equation of the line given the slope equals 5 and the x intercept equals 7?
Call the intercept with the y-axis b.
The equation for the line is then y=5x+b
With the information about the x-intercept we get:
0=5*7+b
b=-35
What four types of operations can be done to functions?
addition, subtraction, multiplication, and division.
What are the steps to fitting equations to data?
- The first step is to determine the nature of the relationship: whether it is linear, quadratic, a higher polynomial, a power function, an inverse, trigonometric, whatever. A scatter plot is a simple way to do this.
- The next step is to determine whether that relationship applies across the whole of the domain or if it changes across it. The extension of a spring is linearly related to the mass hanging from it - until a point where the spring is over-stretched and the relationship (as well as the spring) breaks down. Some systems, particularly those with feedback mechanisms, can become chaotic.
- If necessary, divide up the domain into sub-domains within each of which the relationship is maintained.
- You may wish to consider transformation of the data.
- Then use regression methods to fit a least squares equations. This is an equation that minimises the squares of the differences between the observed values and those given by the equation (the fitted values).
- With a large number of variables you will need to consider removing variables that are correlated with one another. as well as those that contribute little to the regression.
What is the practical application of the study of ergonomics?
Principles of ergonomicsare applied to the design of many elements of everyday life, from car seats to garden tools.
What is the rule for perpindicular lines?
Perpendicular lines intersect at right angles which is 90 degrees
What is the mathematic definition of inequality?
the condition of being unequal; lack of equality; disparity:inequality of size. OR
a statement that two quantities are unequal,indicated by the symbol ≠; alternatively, by the symbol <,signifying that the quantity preceding the symbol is less thanthat following, or by the symbol >, signifying that thequantity preceding the symbol is greater than that following.
Is triple dot product defined?
If by "triple dot product" you mean u·v·w, then no, because that would imply the existence of a dot product between a vector and a scalar.
What is the standard form of linear equation in two variables?
The standard form of a linear equation in two variables, x and y, is
ax + by = k1 where a, b and k1 are constants.
This can be extended to three dimensions (x, y and z) simply:
ax + by + cz = k2 where a, b, c and k2 are constants.
Extension to 4 or more dimensions can be carried out in a similar way.
Apart from the fact that this form lends itself to simple extension to multi-dimensional space, the other main advantage is that the form is easy to represent in matrix form:
Thus AX = K where A is the matrix of coefficients, X the matrix of variables and k the matrix of constants. The tools of matrix algebra can then be used to work with these lines in hyperspace.
The standard form is sometimes confused with the slope-intercept form
y = ax + b.
360
Prime factorization of:
90=2 * 3 * 3 * 5
8 = 2 ...............* 2 * 2
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LCM=2*3 * 3 * 5 * 2 * 2 =
How do you type an exponent when typing on a computer?
To type an exponent when typing on a computer, you can choose of 2 methods: The first method is ** usually inside brackets which makes it useless once you know that The second method is ^
Pythagoras and Plato were some big mathmations in the Greek times, but I don't know how they got started.
