Question #3890

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]

Expert's answer

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.

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.

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