SOLUTION: 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.

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

RELATED QUESTIONS
I don't think it is in the book but here is one I can not solve, please help. Given... (answered by user_dude2008)

Given U = {l, m, n, o, p, q, r, s, t, u, v, w}, A = {l, o, p, q, s, t}, B = {n, o, r, s, (answered by jim_thompson5910)

Use the definition (n) (r) = n!/ r!(n-r)! to prove that (n) = (n ) (r) (answered by Edwin McCravy)

This is what I have for this problem but I cannot figure out (A′ U C′)... (answered by stanbon,jim_thompson5910)

I need help with the following Given U = {l, m, n, o, p, q, r, s, t, u, v, w}, A =... (answered by bcpm36)

Can someone please help me with these two questions, it seems simple, so I am not sure... (answered by jim_thompson5910)

Question 1 please help me with this question. If the universal set U=... (answered by MathLover1)

Let T, U: V to W be linear transformations. Prove that: a. R(T+U) is a subset of... (answered by khwang)

suppose that V is finite-dimensional space and T:V->V is a linear operator. 1. prove... (answered by ikleyn)