# Презентация на тему: Introduction to Vectors

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## Первый слайд презентации: Introduction to Vectors

Karashbayeva Zh.O.

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## Слайд 2: What are Vectors?

Vectors are pairs of a direction and a magnitude. We usually represent a vector with an arrow: The direction of the arrow is the direction of the vector, the length is the magnitude.

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## Слайд 3: Vectors in R n

n = 4 R 4 -space = set of all ordered quadruple of real numbers R 1 -space = set of all real numbers n = 1 n = 2 R 2 -space = set of all ordered pair of real numbers n = 3 R 3 -space = set of all ordered triple of real numbers ( R 1 -space can be represented geometrically by the x -axis) ( R 2 -space can be represented geometrically by the xy -plane) ( R 3 -space can be represented geometrically by the xyz -space)

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## Слайд 4: Multiples of Vectors

Given a real number c, we can multiply a vector by c by multiplying its magnitude by c : v 2 v -2 v Notice that multiplying a vector by a negative real number reverses the direction.

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Two vectors can be added using the Parallelogram Law u v u + v

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## Слайд 6: Combinations

These operations can be combined. u v 2 u - v 2 u - v

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## Слайд 7: Components

To do computations with vectors, we place them in the plane and find their components. v (2,2) (5,6)

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## Слайд 8: Components

The initial point is the tail, the head is the terminal point. The components are obtained by subtracting coordinates of the initial point from those of the terminal point. v (2,2) (5,6)

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## Слайд 9: Components

The first component of v is 5 -2 = 3. The second is 6 -2 = 4. We write v = <3,4> v (2,2) (5,6)

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## Слайд 10: Magnitude

The magnitude of the vector is the length of the segment, it is written || v ||. v (2,2) (5,6)

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## Слайд 11: Scalar Multiplication

Once we have a vector in component form, the arithmetic operations are easy. To multiply a vector by a real number, simply multiply each component by that number. Example: If v = <3,4>, -2 v = <-6,-8>

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To add vectors, simply add their components. For example, if v = <3,4> and w = <-2,5>, then v + w = <1,9>. Other combinations are possible. For example: 4 v – 2 w = <16,6>.

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## Слайд 13: Unit Vectors

A unit vector is a vector with magnitude 1. Given a vector v, we can form a unit vector by multiplying the vector by 1/|| v ||. For example, find the unit vector in the direction <3,4>:

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## Слайд 14: Special Unit Vectors

A vector such as <3,4> can be written as 3<1,0> + 4<0,1>. For this reason, these vectors are given special names: i = <1,0> and j = <0,1>. A vector in component form v = <a,b> can be written a i + b j.

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## Слайд 15

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## Слайд 16

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## Слайд 17: Spanning Sets and Linear Independence

Linear combination :

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## Слайд 18

Ex : Finding a linear combination Sol:

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## Слайд 19

(this system has infinitely many solutions)

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## Слайд 20

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## Слайд 21

If S ={ v 1, v 2,…, v k } is a set of vectors in a vector space V, then the span of S is the set of all linear combinations of the vectors in S, The span of a set: span( S ) Definition of a spanning set of a vector space: If every vector in a given vector space V can be written as a linear combination of vectors in a set S, then S is called a spanning set of the vector space V

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## Слайд 22

Note: The above statement can be expressed as follows Ex 4:

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## Слайд 23

Ex 5: A spanning set for R 3 Sol:

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## Слайд 24

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## Слайд 25

Definitions of Linear Independence (L.I.) and Linear Dependence (L.D.) :

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## Слайд 26

Ex : Testing for linear independence Sol: Determine whether the following set of vectors in R 3 is L.I. or L.D.

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## Слайд 27

EX: Testing for linear independence Determine whether the following set of vectors in P 2 is L.I. or L.D. c 1 v 1 + c 2 v 2 + c 3 v 3 = 0 i.e., c 1 (1+ x – 2 x 2 ) + c 2 (2+5 x – x 2 ) + c 3 ( x + x 2 ) = 0+0 x +0 x 2 c 1 +2 c 2 = 0 c 1 +5 c 2 + c 3 = 0 – 2 c 1 – c 2 + c 3 = 0 Sol: This system has infinitely many solutions (i.e., this system has nontrivial solutions, e.g., c 1 =2, c 2 = – 1, c 3 =3) S is (or v 1, v 2, v 3 are) linearly dependent

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## Слайд 28: Basis and Dimension

Basis : V : a vector space S spans V (i.e., span( S ) = V ) S is linearly independent Spanning Sets Bases Linearly Independent Sets  S is called a basis for V Notes: S ={ v 1, v 2, …, v n } V A basis S must have enough vectors to span V, but not so many vectors that one of them could be written as a linear combination of the other vectors in S V

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## Слайд 29

Notes: (1) the standard basis for R 3 : { i, j, k } i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1) (2) the standard basis for R n : { e 1, e 2, …, e n } e 1 =(1,0,…,0), e 2 =(0,1,…,0),…, e n =(0,0,…,1) Ex: For R 4, {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)}

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## Слайд 30

Ex: matrix space: (3) the standard basis matrix space: (4) the standard basis for P n ( x ): {1, x, x 2, …, x n } Ex: P 3 ( x ) {1, x, x 2, x 3 }

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## Последний слайд презентации: Introduction to Vectors

Ex 2: The nonstandard basis for R 2 Because the coefficient matrix of this system has a nonzero determinant, the system has a unique solution for each u. Thus you can conclude that S spans R 2 Because the coefficient matrix of this system has a nonzero determinant, you know that the system has only the trivial solution. Thus you can conclude that S is linearly independent According to the above two arguments, we can conclude that S is a (nonstandard) basis for R 2

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