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Question 878253: I need help with this question please:
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.
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! Let's start with the definition of the linear map T,
T will satisfy two properties;
1) For all v1 and v2 in R^n , T(v1 + v2) = T(v1) + T(v2)
2) For all v in R^n and λ an element of R, T(λv) = λT(v)
now if L is a line in R^n, then we have the following
R^n = R x R x R .... x R or ordered n-tuples of real numbers
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)
consider two distinct points (x and y) in R^n which define the line
L = (1 - t)x + ty where t is an element of R, then
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
therefore T(L) is a line in R^m
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