Answer to Question #3890 in Linear Algebra for Bob Henry

Question

Suppose M= [1,0;1,1]
Explain why the function T(x)= Mx maps the x-axis onto the line y=x and why it maps the line y=2 onto the line y=x+2 by calculating M[t,0] and where the image points lie and similarily M[t,2]

Answer 1

Suppose M= [1,0;1,1]
Explain why the function T(x)= Mx maps the x-axis onto
the line y=x and
why it maps the line y=2 onto the line y=x+2 by calculating
M[t,0] and
where the image points lie and similarily M[t,2]


***
Remark. The notations of matrix M means that [1,0] is the first row, and [1,1]
is the second row of M


1) Let us show that T maps the x-axis onto the
line y=x.
Let P=[t,0] be any point on the x-axis, then
& T(P) = M[t,0] =
[t,t]
so this point belongs to the line y=x.
Thus the image T(x-axis) is
contained in the line y=x.
Moreover, is Q=[t,t] is any point on y=x, then
Q=T[t,0], so
the image T(x-axis) coincides with the line y=x.

2) Let
us show that T maps the line y=2 onto the line y=x+2.
Similarly Let P=[t,2]
be any point on the line y=2, and T(P)=[X,Y] be the coordinates of its
image.
Then
& T(P) = M[t,2] = [t,t+2] = [X,Y],
so X=t, Y=t+2, and thus
Y=X+2

Thus the image T(line y=2) is contained in the line
y=x+2.
Conversely, if Q=[s,s+2] is any point on the line y=x+2,
Then
Q=T[s,2], so the image T(line y=2) coinsices with the line y=x+2.