www.iPracticeMath.com provides best Math help for every topic.
This presentation shows definition and example of Linear Equation in Two Variables
Welcome to join us at: http://www.facebook.com/iPracticeMath
matrices
The beginnings of matrices goes back to the second century BC although traces can be seen back to the fourth century BC. However it was not until near the end of the 17th Century that the ideas reappeared and development really got underway.
It is not surprising that the beginnings of matrices and determinants should arise through the study of systems of linear equations. The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive.
www.iPracticeMath.com provides best Math help for every topic.
This presentation shows definition and example of Linear Equation in Two Variables
Welcome to join us at: http://www.facebook.com/iPracticeMath
matrices
The beginnings of matrices goes back to the second century BC although traces can be seen back to the fourth century BC. However it was not until near the end of the 17th Century that the ideas reappeared and development really got underway.
It is not surprising that the beginnings of matrices and determinants should arise through the study of systems of linear equations. The Babylonians studied problems which lead to simultaneous linear equations and some of these are preserved in clay tablets which survive.
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This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
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Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
2. 2014/07/31
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7
The Inverse of a Matrix
These entries stretch diagonally down the matrix,
from top left to bottom right.
8
The Inverse of a Matrix
Thus the 2 2, 3 3, and 4 4 identity matrices are
Identity matrices behave like the number 1 in the
sense that
A In = A and In B = B
whenever these products are defined.
9
Example 1 – Identity Matrices
The following matrix products show how multiplying a
matrix by an identity matrix of the appropriate
dimension leaves the matrix unchanged.
10
The Inverse of a Matrix
11
Example 2 – Verifying That a Matrix Is an Inverse
Verify that B is the inverse of A, where
and
Solution:
We perform the matrix multiplications to show that AB = I
and BA = I.
12
Example 2 – Solution cont’d
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13
Finding the Inverse
of a 2 2 Matrix
14
Finding the Inverse of a 2 2 Matrix
The following rule provides a simple way for finding
the inverse of a 2 2 matrix, when it exists.
15
Example 3 – Finding the Inverse of a 2 2 Matrix
Let
Find A–1, and verify that AA–1 = A–1A = I2.
Solution:
Using the rule for the inverse of a 2 2 matrix, we get
16
Example 3 – Solution
To verify that this is indeed the inverse of A, we calculate
AA–1 and A–1A:
cont’d
17
Example 3 – Solution cont’d
18
Finding the Inverse of a 2 2 Matrix
The quantity ad – bc that appears in the rule for
calculating the inverse of a 2 2 matrix is called the
determinant of the matrix.
If the determinant is 0, then the matrix does not have
an inverse (since we cannot divide by 0).
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19
Finding the Inverse
of an n n Matrix
20
Finding the Inverse of an n n Matrix
For 3 3 and larger square matrices the following
technique provides the most efficient way to calculate
their inverses.
If A is an n n matrix, we first construct the n 2n
matrix that has the entries of A on the left and of the
identity matrix In on the right:
21
Finding the Inverse of an n n Matrix
We then use the elementary row operations on this
new large matrix to change the left side into the
identity matrix. (This means that we are changing
the large matrix to reduced row-echelon form.)
The right side is transformed automatically into A–1.
22
Example 4 – Finding the Inverse of a 3 3 Matrix
Let A be the matrix
(a) Find A–1.
(b) Verify that AA–1 = A–1A = I3.
Solution:
(a) We begin with the 3 6 matrix whose left half is A and
whose right half is the identity matrix.
23
Example 4 – Solution
Transform the left half of this new matrix into the identity
matrix by performing the following sequence of elementary
row operations on the entire new matrix.
cont’d
24
Example 4 – Solution
We have now transformed the left half of this matrix into
an identity matrix. (This means that we have put the entire
matrix in reduced row-echelon form.)
Note that to do this in as systematic a fashion as possible,
we first changed the elements below the main diagonal to
zeros, just as we would if we were using Gaussian
elimination.
cont’d
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Example 4 – Solution
We then changed each main diagonal element to a 1 by
multiplying by the appropriate constant(s).
Finally, we completed the process by changing the remaining
entries on the left side to zeros.
The right half is now A–1.
cont’d
26
Example 4 – Solution
We calculate AA–1 and A–1A and verify that both
products give the identity matrix I3.
cont’d
27
Finding the Inverse of an n n Matrix
If we encounter a row of zeros on
the left when trying to find an
inverse, then the original matrix
does not have an inverse.
28
Matrix Equations
29
Matrix Equations
The system
x – 2y – 4z = 7
2x – 3y – 6z = 5
– -3x + 6y + 15z = 0
is equivalent to the matrix equation
30
Matrix Equations
If we let
then this matrix equation can be written as
AX = B
The matrix A is called the coefficient matrix.
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Matrix Equations
We solve this matrix equation by multiplying each side by
the inverse of A (provided that this inverse exists):
AX = B
A–1(AX) = A–1B
(A–1A)X = A–1B
I3 X = A–1B
X = A–1B
Multiply on left by A–1
Associative Property
Property of inverses
Property of identity matrix
32
Matrix Equations
In Example 4 we showed that
So from X = A–1B we have
33
Matrix Equations
Thus x = –11, y = –23, z = 7 is the solution of the original
system.
We have proved that the matrix equation AX = B can be
solved by the following method.
34
Example 6 – Solving a System Using a Matrix Inverse
A system of equations is given.
(a) Write the system of equations as a matrix equation.
(b) Solve the system by solving the matrix equation.
2x – 5y = 15
3x – 6y = 36
35
Example 6(a) – Solution
We write the system as a matrix equation of the form
AX = B.
36
Example 6(b) – Solution
Using the rule for finding the inverse of a 2 2 matrix,
we get
cont’d
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37
Example 6(b) – Solution
Multiplying each side of the matrix equation by this inverse
matrix, we get
So x = 30 and y = 9.
cont’d