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<!DOCTYPE html>
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<title>Subspace, Dimension, Rank</title>
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<h1>Subspace, Dimension, Rank</h1>
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<h2>What is subspace ?</h2>
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<p>A subspace in Rn is a set H in Rn and has the following three properties:</p>
<ol>
<li>1. The zero vector is a member of H;</li>
<li>2. For any vector u and v in H, u+v belongs to H;</li>
<li>3. For any vector u and real Numbers in H, cu belongs to H.</li>
</ol>
<p>That is, the subspace is closed for addition and scalar multiplication. For example, the plane through the origin is a very typical subspace. </p>
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<h2>The basis of a subspace</h2>
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<p>A subspace generally contains an infinite number of vectors, problems in a subspace are best solved by studying a small finite set that can generate the subspace.
A basis for H in Rn is A linearly independent set in H, which generates H, and the parameter vectors for the solution set Ax is equal to 0 are essentially the basis for Null A.</p>
<div align = "center">
<img src="images/subspace.png" height ="100px" >
<img src="images/subspace2.png" height ="120px" >
<h6>{e1 and e2... en} is called the standard basis for Rn.</h6>
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<h2>Dimension of a subspace</h2>
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<p>The dimension of a non-null subspace H is the number of vectors of any basis of H, denoting dim H. The dimension of the null subspace is defined as zero.</p>
<p>Rn has dimensions of n, and each basis of Rn is made up of n vectors, and in R3 a plane that goes through the origin is two-dimensional, and a line that goes through the origin is one-dimensional.</p>
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<h2>Rank of matrix: </h2>
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<p>The rank of matrix A is the dimension of the column space of A, which is called rank A. Since the pivot column of A forms A basis of Col A, the rank of A is exactly the number of pivot column of A after simplifying A to row ladder type.</p>
<p>Determine the rank of A where <img src = "images/rankmatrix.png" height="70px">.</p>
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<img src= "images/rank.png" height="80px">
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<p>Matrix A has three pivot columns, so its rank = 3.</p>
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<h2>Theorem of rank and invertible matrix</h2>
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<p>Assuming that A is an n-by-n matrix, each of the following statements is equivalent to saying that A is invertible.</p>
<ol>
<li>1. The column vectors of A form A basis for Rn;</li>
<li>2. Col A=Rn;</li>
<li>3. dim Col A=n;</li>
<li>4. Rank A=n;</li>
<li>5. Null A= {0};</li>
<li>6. dim Null A=0.</li>
</ol>
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