Let V and W be vector spaces, and let T : V W be a linear transformation. a. Show that T

Problem 8

Let V and W be vector spaces and  be a linear transformation.

(a) Show that T maps the line through u and v to the line T(u) and T(v). What does this mean if  ?

(b) Show that T maps parallel lines to parallel lines.

                                                          Step by Step Solution

Step 1 of 2

Let V and W be vector spaces and  be a linear transformation.

The parametric equation of line joining two vectors u and v is .  

Since T is linear, by definition,

 for any scalars  and  

Now,

 is the equation of line joining  and  

Therefore, T maps the line through u and v to the line T(u) and T(v).

It is seen that .

Now, when , it follows

It implies that T maps the line through u and v to the point T(u)