Linearity is the property of a mathematical relationship (function) that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics include the linear relationship of voltage and current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships are nonlinear.
Generalized for functions in more than one dimension, linearity means the property of a function of being compatible with addition and scaling, also known as the superposition principle.
The word linear comes from Latin linearis, "pertaining to or resembling a line".
The problem is attached. I just wanted to see if the way I proved my statement is correct.
My answer: No, because there exists more columns than rows, thus at least one free variable always exists, thus these vectors are linearly dependent.
Question Source : Elements of Engineering Electromagnetics 6th edition by Rao. Page 202 problem3.30
Problem:
Three sinusoidally time-varying polarized vector fields are given at a point by
F1 = 3^(1/2) * ax * cos(wt +30)
F2 = az * cos(wt+30)
F3 = [ 0.5ax + 3^(1/2)ay + 0.5*3^(1/2)az ] *...
A thin wire of length L and uniformly density ρ is bent into a circular loop with center at O.The moment of inertia of it about a tangential axis lying in the plane of loop is.
Ans : Mass M is not given,but ρ is given. So M=ρL3->(1) (L3 means L cube,no idea how to post it in that manner!). For...
I had kind of a general question. Say I have a second order, homogeneous ODE. Say I use one of the general techniques to generate a complementary solution for my ODE and I end up with something of the form y = C1(solution1) + C2(solution2)
Am I gauranteed that these two solutions will be...
I'm trying to find possible errors in this lab experiment that I did, and one question I am thinking about is "Does ice melt linearly?" I assumed that it does so in my calculations, but now I'm not so sure.
I measured the mass of melted ice (water) over a period of time and assumed that it...
Homework Statement
Suppose that T:W -> W is a linear transformation such that Tm+1 = 0 but Tm ≠ 0. Suppose that {w1, ... , wp} is basis for Tm(W) and Tm(uk) = wk, for 1 ≤ k ≤ p. Prove that {Ti(uk) : 0 ≤ i ≤ m, 1 ≤ j ≤ p} is a linearly independent set.Homework Equations
The Attempt at a Solution...
I have a large real symmetric square matrix (with millions of rows/columns). How can I identify the sets of rows that are linearly dependent?
More generally, can I determine linear independence of rows with a continuous function where, say, the function is 1.0 for a row that is linearly...
The numbers are subscripts.
U1 + U2 + U3 = V1 + V2 + V3
U1 + U2 = V2
I have tried solving for each V in terms of U, but this isn't working out too well.
An example of what I mean:
Suppose you had a blueprint for a chemical rocket.
You build one, and it has mass m and provides thrust x.
Suppose you scale the whole blueprint up by 1% and build another.
The volume (and therefore the mass) of each part in the rocket has increased by a factor of...
Hello everyone! This is my first posting. According to Maxwell, an accelerating charge emits a EM wave. All the books I have referred to, talk about the frequency of oscillating charge. How can we determine the frequency of EM wave emitted by a charge that is accelerating linearly? Thank you...
Hi I am trying to do this problem. Verify that \( y_1=x^3 \) and \(y_2=|x|^3 \) are linearly independent solutions of the diff. equation
\( x^2y''-4xy'+6y=0\) on the interval \((-\infty,\infty) \). Show that \( W(y_1,y_2)=0 \) for every real number x.
I could actually show the above by...
Hi guys,if i mount a dc motor on a glider that slides in 2 directions without friction, AND, instead of having it drive a mechanism, have a circular plate with an eccentrically drilled hole in it (not centre) mounted on the motor shaft though this hole, i should observe the glider sliding back...
Homework Statement
Let S be a linearly independent subset of a Hilbert space. Prove that span(S) is a subspace, that is a linear manifold and a closed set, if and only if S is finite.
Homework Equations
The Attempt at a Solution
Assuming S is finite means that S is a closed set...
Homework Statement
Show that if {a, b, c} is a linearly independent set of vectors, then so are {a, b}, {a, c}, {b, c}, {a}, {b}, and {c}.
Homework Equations
None.
The Attempt at a Solution
Well I was just thinking that if {a, b, c} is a linearly independent set of vectors, then...
Homework Statement
The Attempt at a Solution
I don't think I'm really understanding this problem. Let me tell you what I know: A set is linearly independent if a_1 A_1 +...+a_n A_n = \vec0 for a_1,...,a_n \in R forces a_1 = ...=a_n = 0. If f,g,h take any of the x_i \in S, then one of the...
hallo
I am trying to calculate the probability to obtain 2 sets of linearly independent vectors from a set of binary vectors of length k.
For example:
k = 4, and therefore I have 2^k = 16 vectors to select from.
I want to randomly select 7 vectors (no repetition).
What is the...
Homework Statement
Calculate the total charge embodied in a solid with charge density that decreases linearly with height from a value of λ at the bottom to 0 at the top.
Solve for a rectangular prism and a sphere.
Homework Equations
∫∫∫ρdxdydz
∫∫∫pr^2sinθdrdθd∅
The Attempt at a Solution...
Homework Statement
Let U be the subspace of P3(ℝ) spanned by
E={x^3,x^3-x^2,x^3+x^2,x^3-1}
find a linearly independent subset F of E spanning U.
Homework Equations
E={x^3,x^3-x^2,x^3+x^2,x^3-1}
The Attempt at a Solution
a(x^3)+b(x^3-x^2)+c(x^3+x^2)+d(x^3-1)=0x^3+0x^2+0x+0...
Homework Statement
Let U be the subspace of R5 spanned by the vectors
E={(1,1,0,0,1),(1,1,0,1,1),(0,1,1,1,1),(2,1,-1,0,1)}.
Find a linearly independent subset F of E with Span(E)=U
Homework Equations
The Attempt at a Solution
I figured out that E is linearly dependent and that...
Homework Statement
If the rows of A are linearly dependent, prove that the rows of AB are also linearly dependent.The Attempt at a Solution
A = \begin{pmatrix}a&-a\\b&-b\end{pmatrix} the rows are linearly dependent because a - a = 0 and b - b = 0.
B =...
Finite-dimensional V and W are linearly isomorphic vector spaces over a field. Prove that if \{v_{1},...,v_{n}\} is a basis for V, \{T(v_{1}),...,T(v_{n})\} is a basis for W.
My attempt at a proof:
Let T:V\rightarrow W be an isomorphism and \{v_{1},...,v_{n}\} be a basis for V. Since T is an...
Homework Statement
Using the fact that a set S is linearly dependent if and only if at least one of the vectors, vj, can be expressed as a linear combination of the remaining vectors, obtain necessary and sufficient conditions for a set {u,v} of 2 vectors to be linearly independent. Determine...
Registered events X in time interval t are distributed linearly n = n0 + bt. Find probability density function, then t = 10, n0 = 5 and b =2. Find average amount of registered events per day and Mean squared error. Find the probability to register an event per 5th and 6th days. What is the...
Homework Statement
Let V be a real vector space and {b_1,b_2,b_3,b_4} a linearly independent set of vectors in V
Is the set \left \{ b_1,b_2,b_3,b_1+b_4,b_2+b_4 \right \}
The Attempt at a Solution
\alpha_1b_1+\alpha_2b_2+\alpha_3b_3+\alpha_4\left \{ b_1+b_4 \right \}+\alpha_5\left \{...
So i have 3 vectors:
a= [1 1 1]
b= [2 L 0]
c= [L 2 3]
How do I calculate the L in order to make these vecotrs linearly dependent?
How does ß depend from L if v= [ß 0 -1] and v is in span(a b c)?
Thank you!
Homework Statement
Let F be a subset of the complex numbers. Let V be a vector space over F, and suppose α, β and γ are linearly independent vectors in V. Prove that (α+β), (β+γ) and (γ+α) are linearly independent.
Homework Equations
None.
The Attempt at a Solution
None.
Thanks for your time.
Homework Statement
"In each of the given cases, decide whether the specified elements of the given vector space V (i) are linearly independent, (ii) span V, and (iii) form a basis. Show all reasoning.
V is the space of all infinite sequences (a0, a1, a2, ...) of real numbers v1 =...
Given matrices in a vectorspace, how do you go about determining if they are independent or not?
Since elements in a given vectorspace (like matrices) are vector elements of the space, I think we'd solve this the same way as we've solved for vectors in R1 -- c1u1 + c2u2 + c3u3 = 0. But I'm...
if i have a 4x3 matrix, this means there are more equations than unknowns and so there are no solutions to the system.
does this mean that the row vectors are linearly dependent?
hey i have the set
s = {(t,1,1),(1,t,1),(1,1,t)} and i want to find for which values of t this set is linearly independent.
For a set of vectors containing all numbers i setup c1v1 + c2v2 .. +cnvn = 0 and I need the only solution to be c1=c2=c3..=cn=0 for linear independence.
so then put...
Is a linear equation y'+P(x)y=Q(x) not linear if P(x) and Q(x) are not linearly dependent function?
Does linearly dependent mean a constant multiplied by P(x) will equal Q(x)?
Thank you.
If I'm given a set of vectors
{-4; 3; -10} = v1
{2; -2; -3+k} = v2
{2; -6; 14} = v3
I want to find that they are linearly independent if and only if k != something
to solve this is simple but a huge tedious pain (although not nearly as tedious as trying to find a solution to this...
Homework Statement
A half wave plate and a quarter wave plate are placed between a Polariser and an analyser .All of these are parallel to each other and perpendicular to the direction of propagation of unpolarised incident light.The optic-axis of the half-wave plate makes an angle 300 with...
Hi,
Assume that the real positive numbers x_1,x_2,...,x_n are linearly dependent over the rational numbers, i.e. there are q_1,...,q_n in Q such that x_1*q_1+...+x_n*q_n=0. Is there an algorithm to calculate the coefficients q_i? Is there an algorithm to even check if the x_i's are linearly...
Hey guys was wondering if anyone knew what the go is with linearly dependent solutions to test for exactness, by that I mean
I have the differential equation (2x + y^2)dx + 4xydy = 0 (M,N)
So i test for exactness and
\partialM/\partialy = 2y \partialN/\partialx = 4y
So I...
Homework Statement
I need to argue this properly
Let's say I have a matrix A and rref(A) is given as
\begin{bmatrix}
1 & 0&-1 \\
0& 1 & -1
\end{bmatrix}
Since I have a pivot in every row, why isn't this linearly independent? Don't give me other arguments like "because there...
Homework Statement
Here is a really simple lin.alg problem that for some reason I'm having trouble doing.
Assume that \left\{ v_i \right\} is a set of linearly independent vectors. Take w to be a non-zero vector that can be written as a linear combination of the v_i . Show that \left\{ v_i...
So i have a problem in front of me
Let A be a m x n matrix whose rows are linearly independent. Prove that there exists a vector p such taht Ap = e_1 where e_1 =( 1, 0 , 0, 0, 0,0 ,0 ... 0)T
i don't even know where to begin
Homework Statement
The vectors a, b are linearly independent. For what values of t are = t^2a + b and d = (2t-3)(a-b) linearly independent.
also another similar question
If the vectors a, b , c are linearly independent, show that a-2b-c, 2a+b, and a+b+c are also linearly...
Homework Statement
Basically, the title says it all, I need to figure out whether these functions are linearly independtend on (-infinity, infinity)
Homework Equations
Wronskian (the determinant of the matrix composed of the functions in the first row, first derivative in the second...
Homework Statement
Let u1 = (2; 1; 1; 1) and u2 = (4; 2; 2;-1).I need to extend the linearly independent set u1 and u2 to obtain a basis of R^4.
Homework Equations
The Attempt at a Solution
u1 and u2 are linearly independent since both vectors are non-zero and none is a multiple...
Homework Statement
I need to prove that, if {u;v;w} is a linearly independent set in a
vector space, then the set
{2u + v + w; u + 2v + w; u + v + 2w}
is also linearly independent.
Homework Equations
...
The Attempt at a Solution
if {u;v;w} is a linearly independent set=>...
Homework Statement
V is a subspace of Rn and S={v1,...,vk} is a set of linearly independent vector in V. I have to prove that any list of linearly independent vectors can be extended to a basis for V.
Homework Equations
None that I can think of.
The Attempt at a Solution
So to be...
Homework Statement
Suppose that a matrix A has real entries (which we always assume) and a complex
(non-real) eigenvalue \lambda= a + ib , with b not equal to 0. Let W = U + iV be the corresponding
complex eigenvector, having real and imaginary parts U and V , respectively. Show that
U...
Homework Statement
Show that every m×n matrix A with m linearly independent rows can be obtained
from n × n matrix by deleting the last n − m rows.
Homework Equations
The Attempt at a Solution
I have no idea of this question