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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<title>Vector Equation</title>
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<h1 class="Course_z_off">Linear Algebra</h1>
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<dt>Linear Equation</dt>
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<dt>Vector Space</dt>
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<dt>Eigenvalues</dt>
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<div class="Course_chapter">
<h1>Vector Equation</h1>
<hr color = "#708ACE" size=1px/>
<h2>What is vector ?</h2>
<div class="Course_chapter_content">
<p>Ordered real number pair. A matrix with only one column is called a column vector. E.g.
<img src="images/vector.png" height ="35px" ></p>
<p>The set of vectors of all two elements is called R2, where R means that the members of the vector are real Numbers, and the exponent 2 means that each vector contains two elements.</p>
<p>Geometric representation of R2: since each point on the plane is determined by an ordered pair of real Numbers, the geometric point (a, b) can be equated with the column vector <img src = "images/vector1.png" height="25px">, thus R2 can be regarded as a set of all points on the plane. <br/>
</p>
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<img src="images/plane.png" height ="200px" >
<h6>Write the vector in terms of points</h6>
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<img src="images/plane2.png" height ="200px" >
<h6>Write the vector in terms of arrows</h6>
</div>
<p>Geometric representation of R3: vectors in R3 are 3 by 1 matrices that represent points in the three-dimensional coordinate space,
or arrows that start at the origin.</p>
<div align = "center">
<img src="images/plane3.png" height ="200px" >
<h6>a = <img src = "images/vector2.png" height= "30px"> and 2a</h6>
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<hr color="#708ACE" size=1px/>
<h2>The span of the vector</h2>
<div class="Course_chapter_content">
<p>If v1,v2... vp are vectors in R^n, so the set of all linear combinations of v1,v2... vp are represented by the notation Span{v1,v2... vp},
which is called a subset of R^n generated (or spanned) by v1,v2... vp, that is to say, Span{ v1,v2... vp } is a set of vectors visible as c1v1+c2v2+…+cpvp, where c1,c2…cp is a scalar.</p>
<p>Span{v}: if v is a vector in R^3, then Span{v} is the set of all scalar multiples of v.</p>
<div align = "center">
<img src="images/plane4.png" height ="200px" >
<h6>the set of all the points in R^3 that go through the epsilon and the origin</h6>
</div>
<p>Span{u,v}: if u,v are non-zero vectors in R^3, and v is not a multiple of u, then Span{u,v} is a plane in R^3 through the origin,
including R^3 through u and the origin of the line, including through the origin of the line.</p>
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