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Linear algebra and Vector

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Linear algebra is the study of vectors and linear transformations. Vectors are objects that can be added together and multiplied by scalars. Examples of vectors include numbers, points in space, solutions to linear equations, and rows and columns in tables. Matrices are a special type of linear function that can be used to represent systems of linear equations. Solving systems of linear equations using matrices involves finding the values of variables that simultaneously satisfy all equations in the system.

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By: AnwAr Ali Linear Algebra ? Outline ? What is Linear Algebra? ? What are Vectors? 6 / 1 6 / 2 0 1 6 1
Linear algebra is the study of vectors and linear transformations. In broad terms, vectors are things you can add and linear functions are very special functions of vectors that respect vector addition. To understand this a little better, lets try some examples. WHAT IS LINEAR ALGEBRA? 6 / 1 6 / 2 0 1 6 2
Here are some examples of things that can be added: Example 1 (Vector Addition) (A) Numbers: If x and y are numbers then so is x + y. WHAT ARE VECTORS? 6 / 1 6 / 2 0 1 6 3
WHAT ARE VECTORS? 6 / 1 6 / 2 0 1 6 4
Numbers are not the only things that are vectors, as examples C,D, and E show. Because they can be added", you should now start thinking of all the above objects as vectors! In the above examples, however, notice that the vector addition rule stems from the rules for adding numbers. WHAT ARE VECTORS? 6 / 1 6 / 2 0 1 6 5
When adding the same vector over and over, for example x + x ; x + x + x ; x + x + x + x ; : : : ; we will write 2x ; 3x ; 4x ; : : : ; WHAT ARE VECTORS? 6 / 1 6 / 2 0 1 6 6
WHAT ARE VECTORS? 6 / 1 6 / 2 0 1 6 7
Now the conclusion is that; Vectors are things you can add and scalar multiply. WHAT ARE VECTORS? Thank You 6 / 1 6 / 2 0 1 6 8
? 6 / 1 6 / 2 0 1 6 9
TOPIC MATRICES By: Ghulam Raza 6 / 1 6 / 2 0 1 6 10
Matrices: are linear functions of a certain kind. One way to learn about them is by studying systems of linear equations. Example 4 A room contains x bags and y boxes of fruit: WHAT ARE MATRICES? 6 / 1 6 / 2 0 1 6 11
WHAT ARE MATRICES? 6 / 1 6 / 2 0 1 6 12
Each bag contains 2 apples and 4 bananas and each box contains 6 apples and 8 bananas. There are 20 apples and 28 bananas in the room. Find x and y. The values are the numbers x and y that simultaneously make both of the following equations true: 2 x + 6 y = 20 4 x + 8 y = 28 : WHAT ARE MATRICES? 6 / 1 6 / 2 0 1 6 13
WHAT ARE MATRICES? 6 / 1 6 / 2 0 1 6 14
Thank You 6 / 1 6 / 2 0 1 6 15

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Linear algebra and Vector

  • 1. By: AnwAr Ali Linear Algebra ? Outline ? What is Linear Algebra? ? What are Vectors? 6 / 1 6 / 2 0 1 6 1
  • 2. Linear algebra is the study of vectors and linear transformations. In broad terms, vectors are things you can add and linear functions are very special functions of vectors that respect vector addition. To understand this a little better, lets try some examples. WHAT IS LINEAR ALGEBRA? 6 / 1 6 / 2 0 1 6 2
  • 3. Here are some examples of things that can be added: Example 1 (Vector Addition) (A) Numbers: If x and y are numbers then so is x + y. WHAT ARE VECTORS? 6 / 1 6 / 2 0 1 6 3
  • 4. WHAT ARE VECTORS? 6 / 1 6 / 2 0 1 6 4
  • 5. Numbers are not the only things that are vectors, as examples C,D, and E show. Because they can be added", you should now start thinking of all the above objects as vectors! In the above examples, however, notice that the vector addition rule stems from the rules for adding numbers. WHAT ARE VECTORS? 6 / 1 6 / 2 0 1 6 5
  • 6. When adding the same vector over and over, for example x + x ; x + x + x ; x + x + x + x ; : : : ; we will write 2x ; 3x ; 4x ; : : : ; WHAT ARE VECTORS? 6 / 1 6 / 2 0 1 6 6
  • 7. WHAT ARE VECTORS? 6 / 1 6 / 2 0 1 6 7
  • 8. Now the conclusion is that; Vectors are things you can add and scalar multiply. WHAT ARE VECTORS? Thank You 6 / 1 6 / 2 0 1 6 8
  • 9. ? 6 / 1 6 / 2 0 1 6 9
  • 10. TOPIC MATRICES By: Ghulam Raza 6 / 1 6 / 2 0 1 6 10
  • 11. Matrices: are linear functions of a certain kind. One way to learn about them is by studying systems of linear equations. Example 4 A room contains x bags and y boxes of fruit: WHAT ARE MATRICES? 6 / 1 6 / 2 0 1 6 11
  • 12. WHAT ARE MATRICES? 6 / 1 6 / 2 0 1 6 12
  • 13. Each bag contains 2 apples and 4 bananas and each box contains 6 apples and 8 bananas. There are 20 apples and 28 bananas in the room. Find x and y. The values are the numbers x and y that simultaneously make both of the following equations true: 2 x + 6 y = 20 4 x + 8 y = 28 : WHAT ARE MATRICES? 6 / 1 6 / 2 0 1 6 13
  • 14. WHAT ARE MATRICES? 6 / 1 6 / 2 0 1 6 14
  • 15. Thank You 6 / 1 6 / 2 0 1 6 15