document.write( "Question 356159: I need help with this proof: Let x1....xk be linearly independent vectors in R^n, and let A be a nonsingular nXn matrix. Define yi = Axi for i = 1,...k. show that y1,.... yk are linearly independent. Thanks! \n" ); document.write( "

Algebra.Com's Answer #254231 by robertb(5830) $\"\"$ $\"About$

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Consider the linear combination:
\n" ); document.write( "(a1)(y1)+(a2)(y2)+(a3)(y3)+....+(a(n-1))(y(n-1))+(an)(yn)= 0, a1, a2, a3,...an are scalar coefficients.
\n" ); document.write( "For the purpose of contradiction, suppose that {y1, y2, y3,...,yn} is a linearly dependent set. Therefore not all of a1, a2, a3, ...an are equal to zero, by definition. Since A is nonsingular, $\"A%5E-1\"$ exists.
\n" ); document.write( "Now
\n" ); document.write( "(a1)(y1)+(a2)(y2)+(a3)(y3)+....+(a(n-1))(y(n-1))+(an)(yn)=
\n" ); document.write( "(a1)(Ax1)+(a2)(Ax2)+(a3)(Ax3)+....+(a(n-1))(Ax(n-1))+(an)(Axn)=
\n" ); document.write( "A((a1)(x1)+(a2)(x2)+(a3)(x3)+....+(a(n-1))(x(n-1))+(an)(xn))= 0.
\n" ); document.write( "Since A is nonsingular, this means that $\"A%5E1\"$ exists. Left-multiply the equation
\n" ); document.write( "A((a1)(x1)+(a2)(x2)+(a3)(x3)+....+(a(n-1))(x(n-1))+(an)(xn))= 0
\n" ); document.write( "by $\"A%5E-1\"$ . This means that
\n" ); document.write( "(a1)(x1)+(a2)(x2)+(a3)(x3)+....+(a(n-1))(x(n-1))+(an)(xn)= 0,and
\n" ); document.write( "not all a1, a2, a3, ...an are equal to zero, CONTRADICTION, because {x1, x2, x3, ...xn} is a linearly independent set. Therefore
\n" ); document.write( "{y1, y2, y3,...,yn} has to be a linearly independent set.
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