The document discusses linear independence and change of basis in vector spaces. It provides the following key points:
1) Two vectors u and v are linearly dependent if one is a multiple of the other, and independent otherwise.
2) Three vectors u, v, and w are linearly dependent if their coefficients in a linear combination equal 0, and independent otherwise.
3) A change of basis matrix P describes the transformation between two bases {e} and {f} of a vector space, where each vector in {f} is written as a linear combination of the vectors in {e}. The inverse of P transforms vectors back from {f} to {e}.
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International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Fixed Point Results for Weakly Compatible Mappings in Convex G-Metric Spaceinventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
In this chapter, the authors extend the theory of
the generalized difference Operator ∆L to the generalized
difference operator of the 풏
풕풉kind denoted by ∆L Where L
=푳 = {풍ퟏ,풍ퟐ,….풍풏} of positive reals풍ퟏ,풍ퟐ,….풍풏and obtain some
interesting results on the relation between the generalized
polynomial factorial of the first kind, 풏
풕풉kind and algebraic
polynomials. Also formulae for the sum of the general
partial sums of products of several powers of consecutive
terms of an Arithmetic progression in number theory are
derived.
Exact Solutions of the Klein-Gordon Equation for the Q-Deformed Morse Potenti...ijrap
In this work, we solve the Klein-Gordon (KG) equation for the general deformed Morse potential with
equal scalar and vector potentials by using the Nikiforov-Uvarov (NU) method, which is based on the
solutions of general second-order linear differential equation with special functions. The energy
eigenvalues and corresponding normalized eigenfunctions are obtained. It is found that the eigenfunctions
can be expressed by the Laguerre polynomials. Our solutions have a good agreement with earlier study.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Flip bifurcation and chaos control in discrete-time Prey-predator model irjes
The dynamics of discrete-time prey-predator model are investigated. The result indicates that the
model undergo a flip bifurcation which found by using center manifold theorem and bifurcation theory.
Numerical simulation not only illustrate our results, but also exhibit the complex dynamic behavior, such as the
periodic doubling in period-2, -4 -8, quasi- periodic orbits and chaotic set. Finally, the feedback control method
is used to stabilize chaotic orbits at an unstable interior point.
In this chapter, the authors extend the theory of
the generalized difference Operator ∆L to the generalized
difference operator of the 풏
풕풉kind denoted by ∆L Where L
=푳 = {풍ퟏ,풍ퟐ,….풍풏} of positive reals풍ퟏ,풍ퟐ,….풍풏and obtain some
interesting results on the relation between the generalized
polynomial factorial of the first kind, 풏
풕풉kind and algebraic
polynomials. Also formulae for the sum of the general
partial sums of products of several powers of consecutive
terms of an Arithmetic progression in number theory are
derived.
Exact Solutions of the Klein-Gordon Equation for the Q-Deformed Morse Potenti...ijrap
In this work, we solve the Klein-Gordon (KG) equation for the general deformed Morse potential with
equal scalar and vector potentials by using the Nikiforov-Uvarov (NU) method, which is based on the
solutions of general second-order linear differential equation with special functions. The energy
eigenvalues and corresponding normalized eigenfunctions are obtained. It is found that the eigenfunctions
can be expressed by the Laguerre polynomials. Our solutions have a good agreement with earlier study.
International Journal of Mathematics and Statistics Invention (IJMSI)inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Flip bifurcation and chaos control in discrete-time Prey-predator model irjes
The dynamics of discrete-time prey-predator model are investigated. The result indicates that the
model undergo a flip bifurcation which found by using center manifold theorem and bifurcation theory.
Numerical simulation not only illustrate our results, but also exhibit the complex dynamic behavior, such as the
periodic doubling in period-2, -4 -8, quasi- periodic orbits and chaotic set. Finally, the feedback control method
is used to stabilize chaotic orbits at an unstable interior point.
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This will be used as part of your Personal Professional Portfolio once graded.
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Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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1. NPTEL – Physics – Mathematical Physics - 1
Lecture 9
Linear independence
Determine whether u and v are linearly independent
i) 𝑢 = (1, 2), 𝑣 = (3, −5), (ii) 𝑢 = (1, −3), 𝑣 = (−2𝑢)
2 vectors are said to be linearly dependent if one is multiple of another.
a)
b)
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u and v are independent
u and v are dependent for 𝑣 = −2𝑢
Determine whether 3 vectors
𝑢 = (1, 1, 2), 𝑣 = (2, 3, 1) and 𝑤 = (4, 5, 5) are linearly independent.
1 2 4 0
𝑥 [1] + 𝑦 [3] + 𝑧 [5] = [0]
2 1 5 0
If this set has a non-zero solution for (x, y, z) then they are linearly dependent.
The students should check this.
Change of basis
Let {𝑒1, 𝑒2 … … … 𝑒𝑛} is a basis of a vector space v and {𝑓1 … … … … 𝑓𝑛} is another
basis. Suppose there is a relation that exists between the two bases such that,
𝑓1 = 𝑎11𝑒1 + 𝑎12𝑒2 + … … … 𝑎1𝑛𝑒𝑛
𝑓2 = 𝑎21𝑒1 + 𝑎22𝑒2 + … … … 𝑎2𝑛𝑒𝑛
-
-
-
-
-
-
-
-
-
𝑓𝑛 = 𝑎𝑛1𝑒1 + 𝑎𝑛2𝑒2 + … … … 𝑎𝑛𝑛𝑒𝑛
Then the transpose, P of the above matrix of coefficients is called the basis
matrix.
Theorem 1
Let P be a basis matrix from a basis {e;} to a basis {𝑓𝑗 } and Q be the change of
basis matrix from the basis {𝑓𝑗 } to {𝑒𝑖} back. Then P is invertible and 𝑄 = 𝑃−1
Proof 𝑓𝑖 = ∑𝑛 𝑎𝑖𝑗 𝑒
𝑗 =1
𝑗
(1)
(2)
Also 𝑒𝑖 = ∑𝑛
𝑘=1 𝑏𝑗 𝑘 𝑓
𝑘
Substituting (2) in (1)
𝑓𝑖 = ∑𝑛 𝑎𝑖𝑗(∑𝑛 𝑏𝑗𝑘𝑓𝑘 ) = ∑𝑛 (∑𝑛 𝑎 𝑏 ) 𝑓
𝑗 =1 𝑘=1 𝑘=1 𝑗 =1 𝑖 𝑗 𝑗 𝑘 𝑘
Now, ∑𝑗 𝑎𝑖𝑗𝑏𝑗𝑘 = 𝛿𝑖𝑘
2. NPTEL – Physics – Mathematical Physics - 1
Where 𝛿𝑖𝑘 is the Kronecker delta function with the following properties,
𝛿𝑖𝑘 = 1 for 𝑖 = 𝑘
= 0 for i≠ 𝑘 So, 𝐶𝑖𝑘 = 𝛿𝑖𝑘 so 𝑃𝑄 = 1 Or 𝑃 = 𝑄−1 (proved)
Example
Consider the following bases in 𝑅2.
𝑆1 = {𝑢1 = (1, −2), 𝑢2 = (3, −4)}
𝑆2 = {𝑣1 = (1,3), 𝑣2 = (3,8)}
(i) Find the components of an arbitrary vector (𝑏) in 𝑅2 in basis 𝑆1 = {𝑢1, 𝑢2}.
(ii) Write the change of basis matrix P from 𝑆1to 𝑆2. To do this we have to write
𝑣1 and 𝑣2 in terms of 𝑢1 and 𝑢2.
𝑎
Solution
(𝑏) = 𝑥 ( ) + 𝑦 ( ) ⇒ 𝑥 + 3𝑦 = 𝑎 and −2𝑥 − 4𝑦 = 𝑏
𝑎 1
−2 −4
3
3 1
Thus, (𝑎1𝑏)𝑠1 = (−2𝑎
− 2
𝑏) 𝑢1 + (𝑎 + 2
𝑏)
𝑢2
Joint initiative of IITs and IISc – Funded by MHRD Page 10 of 28