If v notequalto 0 and p are vectors in Rn the line e through

If v notequalto 0 and p are vectors in R^n, the line e through p in the direction of y has the parametric equation x = p + tv, t element R. Suppose T: R^n rightarrow R^m is a linear transformation. Show that T maps e onto another line or onto a single point (a degenerate line).

Solution

Let T be defined as y = T(x). Then y = T(x) = T(p+tv) = T(p) +tT(v) ( as T is a linear transformation). Now, if T(v) 0, then y describes a line passing through T(p) in the direction of the vector T(v). Further,in case T(v) = 0, then y is a single point (degenerate line).