document.write( "Question 878253: I need help with this question please:\r
\n" ); document.write( "\n" ); document.write( "Given that T is a linear map from R^n to R^m and L is a line in R^n, prove that T(L) is also a line. \n" ); document.write( "

Algebra.Com's Answer #529855 by rothauserc(4718) $\"\"$ $\"About$

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Let's start with the definition of the linear map T,
\n" ); document.write( "T will satisfy two properties;
\n" ); document.write( "1) For all v1 and v2 in R^n , T(v1 + v2) = T(v1) + T(v2)
\n" ); document.write( "2) For all v in R^n and λ an element of R, T(λv) = λT(v)
\n" ); document.write( "now if L is a line in R^n, then we have the following
\n" ); document.write( "R^n = R x R x R .... x R or ordered n-tuples of real numbers
\n" ); document.write( "L in R^n says L = (x1, x2, x3, ....,xn) and (x1, x2, x3, ..., xn) are called the coordinates of L which is a vector (directed line)
\n" ); document.write( "consider two distinct points (x and y) in R^n which define the line
\n" ); document.write( "L = (1 - t)x + ty where t is an element of R, then
\n" ); document.write( "T(L) = T((1-t)x +ty) = T((1-t)x) + T(ty) = (1-t)T(x) + tT(y) using properties of 1 and 2
\n" ); document.write( "therefore T(L) is a line in R^m\r
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