Презентация на тему: Introduction to Vectors

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Introduction to Vectors
What are Vectors?
Vectors in R n
Multiples of Vectors
Adding Vectors
Combinations
Components
Components
Components
Magnitude
Scalar Multiplication
Addition
Unit Vectors
Special Unit Vectors
Introduction to Vectors
Introduction to Vectors
Spanning Sets and Linear Independence
Introduction to Vectors
Introduction to Vectors
Introduction to Vectors
Introduction to Vectors
Introduction to Vectors
Introduction to Vectors
Introduction to Vectors
Introduction to Vectors
Introduction to Vectors
Introduction to Vectors
Basis and Dimension
Introduction to Vectors
Introduction to Vectors
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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Слайд 5: Adding Vectors

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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Слайд 12: Addition

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

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Ex 5: A spanning set for R 3 Sol:

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Слайд 24

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

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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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