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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 does it mean if there are an infinite number of solutions to a system of linear equations?
Any two numbers that make one of the equations true will make the other equation true.
What are parent functions in math?
They are basic functions on which other functions depend.
for ex. x^2 is a parent function for 2x^2.
What are linear equtions and simultanions equations?
Linear Equations are equations with variable with power 1
for eg:
5x + 7 = 0
Simultaneous Equations are two equations with more than one variable so that solving them simultaneously
What are the LACSAP fractions?
The LACSAP fractions are a set of fractions set in a geometric pattern that are part of one of the two portfolio any International Baccalaureate - Diploma student must complete.
How do you know if it is a linear equation?
An equation is linear if and only if all of the unknown quantities are multiplied by a constant and added to the rest of the left or right side. In other words, there are no powers, trig, etc.
ex.
5x -y = 12 is linear.
3a +4b +5c +19d +pi*f = 0 is linear.
sin(x) = 0 is not.
x^2 = y-15 is not.
How do you find the equation that matches a scattered plot?
use a graphing calculator. chances are, it's not linear; it's probably got some high-degree function with non-whole number coefficients. i think you can download a free graphing calculator online.
What is an arrangement of items in which the order of the items is not considered?
Multiplication (3 x 5 = 15, and 5 x 3 = 15), or Addition (5 + 3 = 8, 3 + 5 = 8).
What do you get if round to nearest tenth 5.19?
5.19 rounded to the nearest tenth is 5.20.
The first number after the decimal point (in 5.19 it is 1) is the tenths place; the number after that is the hundredths place. If the number in the hundredths place is 4 or below, then the tenths place simply remains the same, but if the hundredths place is 5 or above, then the tenths place is rounded up to the next number.
A diagonal forms two non-congruent triangles?
False. A diagonal of a parallelogram produces 2 congruent triangles
What are the zeros of the function f of x equals 5x cubed minus x squared minus 18x plus 8?
-2, 1.74 and 0.46
Draw a flow chart for finding roots of quadratic equation?
START
INPUT a,b,c
d=B*B-4ac
x1=-b+d/2a
x2=-b-d/2a
print x1,x2
stop
What number multiplied by itself is 625?
If you listen you what you saying, you're simply asking for √625.
The answer is 25.
* * * * *
That is one of the two possible answers. -25 is also an answer.
say x= the amount of water in one tank
x+3000=6(x-3000) (you subtract 3000 from one tank and give it to the other)
x+3000=6x-18000
21000=5x
21000/5=x
4200=x
I'm not going to lie, I'm not 100% sure if I'm going to do this correctly. It's been a while since I've done something like this, so you may want to double check my answer. Also, the letter T in your question is really throwing me off, so I'm just going to give you two answers. The first answer will treat T as a variable, the second answer will ignore it completely. Both answers, however, use the following equation for the reflection of a vector about a line:
RefL(v) = 2L(v â— L)/(L â— L) - v
For my first answer, I'll use the following vectors for Land v:
L = -2i - j + 2Tk, and
v = 7i + 2j + 7Tk,
where i, j, and k are the unit vectors in the direction of the x, y, and z axes in R3, respectively.
Thus,
v â— L = -14 - 2 + 14T2 = 14T2 - 16.
L â— L = 4 + 1 + 4T2 = 4T2 + 5.
Therefore, 2L(v â— L)/(L â— L) =
2L(14T2 - 16)/(4T2 + 5) = L(28T2 - 32)/(4T2 + 5) =
-8(7T2 - 8)/(4T2 + 5)i - 4(7T2 - 8)/(4T2 + 5)j + 8T(7T2 - 8)/(4T2 + 5)k.
Let A = 2L(v â— L)/(L â— L).
(A - v)i = [-8(7T2 - 8)/(4T2 + 5) - 7(4T2 + 5)/(4T2 + 5)]i =
(-56T2 + 64 - 28T2 - 35)/(4T2 + 5)i = (-84T2 + 29)/(4T2 + 5)i.
(A - v)j = [-4(7T2 - 8)/(4T2 + 5) - 2(4T2 + 5)/(4T2 + 5)]j =
(-28T2 + 32 - 8T2 - 10)/(4T2 + 5)j = (-36T2 + 22)/(4T2 + 5)j.
(A - v)k = [8T(7T2 - 8)/(4T2 + 5) - 7T(4T2 + 5)/(4T2 + 5)]k =
(56T3 - 64T - 28T3 - 35T)/(4T2 + 5)k = (28T3 - 99T)/(4T2 + 5)k.
Let b = 1/(4T2 + 5), then
RefL(v) = b[(-84T2 + 29)i + (-36T2 + 22)j + (28T3 - 99T)k]
That expression for RefL(v) looks pretty ugly, so I'm going to do the problem again, this time without the variable T.
L = -2i - j + 2k, and
v = 7i + 2j + 7k.
v â— L = -14 - 2 + 14 = -2
L â— L = 4 + 1 + 4 = 9
Therefore, 2L(v â— L)/(L â— L) =
2L(-2/9) = L(-4/9) =
(8/9)i + (4/9)j - (8/9)k.
RefL(v) = 2L(v â— L)/(L â— L) - v = (8/9)i + (4/9)j - (8/9)k - 7i - 2j - 7k =
-(55/9)i - (14/9)j - (71/9)k.
While this expression does look much nicer, I'm not sure if it's right. So, like I recommended above, please double check my work!
What is the difference between algebra and linear algebra?
Linear Algebra is a special "subset" of algebra in which they only take care of the very basic linear transformations. There are many many transformations in Algebra, linear algebra only concentrate on the linear ones.
We say a transformation T: A --> B is linear over field F if
T(a + b) = T(a) + T(b) and kT(a) = T(ka)
where a, b is in A, k is in F, T(a) and T(b) is in B. A, B are two vector spaces.
What is consistent and dependent?
The terms consistent and dependent are two ways to describe a system of linear equations. A system of linear equations is dependent if you can algebraically derive one of the equations from one or more of the other equations. A system of linear equations is consistent if they have a common solution.
An example of a dependent system of linear equations:
2x + 4y = 8
4x + 8y = 16
Solve the first equation for x:
x = 4 - 2y
Plug that value of x into the second equation:
16 - 8y + 8y = 16, which gives 16 = 16.
No new information was gained from the second equation, because we already knew 16 = 16, so these two equations are dependent.
An example of an inconsistent system of linear equations:
Because consistency is boring.
2x + 4y = 8
4x + 8y = 15
Solve the first equation for x:
x = 4 - 2y
Plug that value of x into the second equation:
16 - 8y + 8y = 15, which gives 16 = 15.
This is a contradiction, because 16 doesn't equal 15. Therefore this system has no solution and is inconsistent.
Since the columns of AT equal the rows of A by definition, they also span the same space, so yes, they are equivalent.
How is solving radical equations similar to solving linear equations?
It really is utilized to solve specific variables
It really is utilized to rearrange the word.
