Algebra.Com's Answer #836056 by ikleyn(52781)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "hello. this is from my 'spanning sets' lesson from vectors:
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document.write( "the set of vectors {(1,0,0), (0,1,0)} spans a set in R3
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document.write( "a. describe the set
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document.write( "b. write the vector (-2, 4, 0) as a linear combination of these vectors
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document.write( "c. explain why it is not possible to write ( 3,5,8) as a linear combination of these vectors
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document.write( "d. If we added the vector (1,1,0) to this set, would it now span R3? Explain.
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document.write( " To solve the problem, I will introduce designations for these basic vectors\r
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document.write( " e1 = (1,0,0), e2 = (0,1,0).\r
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document.write( "(a) The spanned set in  is the set of all triples (vectors/points) {(x,y,0)},\r\n" );
document.write( " where the last third component/coordinate is 0 (zero).\r\n" );
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document.write( " You may think about this set as about a plane in defined by equation z= 0.\r\n" );
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document.write( "(b) (-2,4,0) = (-2)*e1 + 4*e2.\r\n" );
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document.write( " This equality is from the area of OBVIOUS truths.\r\n" );
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document.write( " You may check it by using the rules of multiplying a vector by a number\r\n" );
document.write( " and the rules of addition vectors.\r\n" );
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document.write( "(c) It is not possible to write (3,5,8) as a linear combination of these given vectors\r\n" );
document.write( " e1 and e2, because each and every such combination will have the third coordinate 0 (zero),\r\n" );
document.write( " while the given vector (3,5,8) has the third coordinate 8 (not zero).\r\n" );
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document.write( "(d) The answer is NO.\r\n" );
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document.write( " The answer is NO, because the expanded set of basic vectors will still produce zero third coordinate\r\n" );
document.write( " in any linear combination.\r\n" );
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document.write( " Adding this vector (1,1,0) to e1 and e2 does not change the spanned set: it remains the plane z= 0.\r\n" );
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document.write( "Solved, with all necessary explanations.\r
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