document.write( "Question 1160970: Let v=[4, -9, 1, 9]. Find a basis of the subspace of R^4 consisting of all vectors perpendicular to v. \n" ); document.write( "

Algebra.Com's Answer #785147 by rothauserc(4718) $\"\"$ $\"About$

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v = [4, -9, 1, 9]\r
\n" ); document.write( "\n" ); document.write( "Let u be a vector in R^4 and let R^4 be the set of 4 by 1 column vectors
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\n" ); document.write( "Then let W = { u an element of R^4 such that vu = 0 }
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\n" ); document.write( "Show subspace requirements are satisfied, that is, zero vector in R^4 is in W and W is closed under addition and scaler multiplication.
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\n" ); document.write( "The zero element in R^4 is 0, the 4 by 1 column vector whose entries are all 0, then v0 = 0, therefore 0 is an element in W
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\n" ); document.write( "Suppose u, w are elements of W and c is an element of R, then vu = vw = 0 and
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\n" ); document.write( "v(u +w) = vu +vw = 0, therefore u +w is an element in W
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\n" ); document.write( "Since vu = 0, v(cu) = cvu = c0 = 0
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\n" ); document.write( "Vectors u, w in R^4 are said to be perpendicular if u^Tw = 0
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\n" ); document.write( "let v = u^T and we have show the set of vectors perpendicular to any given vector is a subspace of R^4
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