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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
Can a linear programming problem have two optimal solutions?
No. However, a special subset of such problems: integer programming, can have two optimal solutions.
Why do you isolate the variable on one side the the equation when solving a linear equation?
Isolating a single variable in terms of the rest of the equation provides a solution to that variable. That is, if you know the equation that equals the variable, then you can figure out its value.
What is the answer to y equals 3x 3x plus 4y equals 30?
y = 3x
3x + 4y = 30
3x + 4(3x) = 30
3x + 12x = 30
15x = 30
x = 2
y = 3x
y = 3(2)
y = 6
(2, 6)
How do you translate ordered pairs into an equation?
If you have a pair of coordinates you can find the slope then put it into either point slope or slope intercept form.
(2,3) and (5,4) has a slope of (3-2)/2-4) or 1/-2 or -1/2
Then you can put that in y-3=-1/2(x-2) as point slope
or y=-1/2(x) + b
3=-1/2(2) + b
4=b therefore y=-1/2x+4
If you take a calculator and start dividing by numbers 1,2,3 and so on you will get some that give whole answers. These are the ones you are looking for.
1x312
2x156
3x104
4x78
6x52
8x39
13x24
Can linear equation not be rewritten in slope-intercept form?
A linear equation can be written in many different forms. Two forms are used frequently. ax+by=c is standard form as y=mx+b is slope intercept form.
How do you find the x-intercept and y-intercept for 4x plus 9y equals 18?
The x-intercept would be 4x + 9(0)= 18 so x = 4.5 (4.5,0)
the y-intercept you'd make the x=0.
4(0) + 9y = 18 so y=2 (0,2)
How do you solve Linear Equations with the substitution method?
You just plug in the value of the given variable. For example:
y=3+2
2y=x (now you substitute y for 3+2)
2(3+2)=x (now solve the equation using distributive property)
6+4=x
10=x
Tuhduh!! All done using the substitution property.
How do you round 7005.9 to the nearest one?
7,006.
If the place you are converting is < 5, then round down (7,005.49, for example, would round down to 7,005). If the place is ≥ 5, round up.
y = (2/3 x) - 16
What number goes into 26 and 18?
The only common factors I can see are 1 and 2. 18 is divisible by 1,2,3,6, and 9, and 26 is divisible by 1,2,13,and 26.
How do you find inverse of the matrix using augmented matrix?
your first step is to augment(connect with the bar in between) the identity matrix onto the right side of the original matrix
then your main objective is to turn your original matrix into the identity matrix while only using elementary row operations
elementary row operation rules:
- row swap- you can swap any row with another row. meaning the entire first row can be swapped with the second, or third row but only entire rows may be swapped
- row multiplication- you can multiply or divide entire rows by a constant ex: if row1=3t-5s then row1 x 2=6t-10s (note you must multiply every term in the variable)
- row addition- any row can be added or subtracted onto another row to change that row. note if you were to add row1 to row2 for example change would only be done to row2 not row1. (very important)- be careful when adding rows remember that when rows are being added/subtracted, they subtract according to how columns within rows align with. this is where the most mistakes happen.
- (most important)- what ever action you perform on a row you do it to the entire row. you must treat both sides of the line as ONE UNIT(thasshow i remembered). when you perform an operation on the original matrix, you cannot forget to do the same to the augmented peice
if the original matrix is sucessfully changed into the identity matrix using these rules, then you should end up with the identity matrix on the original side and theinverse matrix on the augmented side
examples of this method can be found by looking up linear algebra on khanacadamy.org
How do you write an equation for a line that passes through point and is parallel to the given line?
Both straight line equations will have the same slope or gradient but the y intercepts wll be different
What number is divisible by 2700?
All multiples of 2,700 which is an infinite number. The first five are: 2700, 5400, 8100, 10800, 13500.
What does terms of an expression mean?
An expression is a collection of numbers and variables, along with mathematical operations, but without an equality (or inequality) symbol.
What distinguishes a weak matrix from strong matrix?
In organizational management, a strong matrix is an organizational structure arranged around projects; a weak matrix is arranged around functional roles. For example, in a strong matrix structure, the resources might be organized to support Product A or Product B, in a weak matrix structure, the resources might be organized into Development or Manufacturing.
How does the graph of the line change as you make b larger but keep m the same?
The equation of a line is y = mx + b. If the slope of the line (m) stays the same, the line will be parallel to the original line. What changing b does is change the y-intercept of the line, because when you make x = 0, y = b. So by making b larger, you are moving the line up the y axis.
