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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
Advanced engineering mathematics by erwin kreyszig 9th edition solution?
A book to introduce engineering and physics students to areas of math that seem to be most important in relation to practical problems. Book was first published in 1962 - so it is a bit out of date - and has had several reprints. Erwin Kreyszig (Jan 6, 1922 - December 12, 2008) was Professor of at Ohio State University, later moved to Carleton University in Ottawa).
The book covers: Ordinary Differential Equations; Ordinary Linear Differential Equations; Power Series Solutions of Diff. Equations; Laplace Transform; Vector Analysis; Line and Surface Integrals; Systems of Linear Equations; Fourier Series and Integrals; Partial Differential Equations; Complex analytic Functions; Conformal Mapping; Complex Integrals; and so on. A very useful book when I did my engineering, though it must be out of date now.
GSC
How do you solve for linear system equations?
Simple systems of linear equations involve two equations and two variables. Graphically this may be represented by the intersection of lines in a plane. If the two equations describe the same line or parallel lines, there is no solution.
Example:
x + y = 7
2x - y = 8
We might rewrite the first equation as x = 7 - y (subtracting y from each side).
Then we can substitute 7-y for x in the second equation:
2(7-y) - y = 8
By the distributive property of multiplication over addition this yields:
14 - 2y - y = 8
14 - 3y = 8 (combining -2y and -y)
14 = 8 + 3y (add 3y to each side)
6 = 3y (subtract 8 from each side)
2 = y (divide each side by 2).
If y = 2, we can substitute this back into either equation. The first looks like it would be the easiest: x + 2 = 7.
x is therefore 5.
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.
How are the inverse matrix and identity matrix related?
If an identity matrix is the answer to a problem under matrix multiplication, then each of the two matrices is an inverse matrix of the other.
What is the y intercept in this equation 5x-2y equals 20?
5x-2y=20
Subtract 5x on both sides to see what y equals.
-2y=20-5x
Divide by -2 on each side.
y=-10+(5/2)x or
y = (5/2)x - 10 (the slope-intercept form of the equation of a line, where the y-intercept is -10)
Add 2y to both sides to see what x equals.
5x=20+2y
Divide by 5 on each side.
x=4+2/5y or
x = (2/5)y + 4, where 4 is the x-intercept.
(all this work is to show that when the value of y is zero, the line cuts the x-axis at 4)
Now plug in what x equals into the equation of what y equals to get the exact value of y.
y=-10+5/2x
y=-10+5/2(4+2/5y)
Distribute the 5/2 to the (4+2/5y) in parentheses.
y=-10+10+y
Combine like terms.
y=y+0
Subtract y on both sides.
0y=0
Divide by 0 on each side.
y=0
Now plug in what y equals to into what x equals to find the exact value of x.
x=4+2/5y
x=4+2/5(0)
x=4+0
x=4
Your first solution to the data is (4,0).
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.
Can a component of a vector be greater than the vector itself?
No, the magnitude of a vector (in Euclidean space) is the square root of the sum of the squares of its components. This value can never be greater than the value of one of its own components.
v = √((vx)2 + (vy)2 + (vz)2)
v2 = (vx)2 + (vy)2 + (vz)2
(vx)2 = -(vy)2 - (vz)2 + v2
vx = √(-(vy)2 - (vz)2 + v2)
Can vx > v?
Substituting:
√(-(vy)2 - (vz)2 + v2) > √((vx)2 + (vy)2 + (vz)2).
Simplified:
v2 > (vx)2 + 2(vy)2 + 2(vz)2.
Substituting again:
(vx)2 + (vy)2 + (vz)2 > (vx)2 + 2(vy)2 + 2(vz)2.
Simplifying again:
0 > (vy)2 + (vz)2.
This results in a fallacy, since 0 can't be greater than a positive number. This wouldn't work even if both vy and vz were 0.
How do you solve equations with exponents with y?
Do you mean y=x^2.5? I you had y=13^2.77, it's easier to use log.
log y=2.77*log13 ~ 3.0856. 1217.9 is the antilog and answer.x=1217.9
But math can be more complicated. How about y^2.5=x^1.8. Logs really shine here. Take log of both sides. 2.5*log y = 1.8 log x. Say x=100 and 1.8 log 100 = 1.8*2=3.6. We have 2.5 log y = 3.6 and log y = 3.6/2.5 = 1.44. Now y = antilog 1.44=27.54229. So does 27.54229^2.5 = 100^1.8 ? Yes it does.
How do you solve a mathecrostics?
It's a logical puzzle, make up answere that fit in the open square so it matches the equation on the sides and or bottom.
What equation is equivalent to 3x-4y equals 8?
3x - 4y = 8
Solve for x:
3x = 4y + 8
x = 4/3y + 8/3
Solve for y:
-4y = -3x + 8
y = 3/4x - 2
How would you factor 24x3 plus 72x2 plus 30x?
24x^3 + 72x2 + 30x = 6x*(4x^2 + 12x + 5)=6x*(2x + 1)*(2x + 5)
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 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!
