This linear transformation maps the point (2,1) to...

[Fellowroot]

Homework Statement

Let T:R->R^2 be the linear transformation that maps the point (1,2) to (2,3) and the point (-1,2) to (2,-3). Then T maps the point (2,1) to ...

Homework Equations

T(xa+yb) = xT(a)+yT(b)

The Attempt at a Solution

Okay so I have the solution to this problem, but its understanding some multiplication that's getting me.

They get x=5/4 and y = -3/4

and they do the following

T(c) = xT(a) +yT(b)

T(²₁)=(5/4)T(¹₂)-(3/4)T(^-1₂)

(5/4)(²₃)-(3/4)(²_-3)

(¹₆)

I just need someone to explain to me how they got the 1 and 6 at the end.

 

[Mark44]

Homework Statement

Let T:R->R^2 be the linear transformation that maps the point (1,2) to (2,3) and the point (-1,2) to (2,-3). Then T maps the point (2,1) to ...

Homework Equations

T(xa+yb) = xT(a)+yT(b)

The Attempt at a Solution

Okay so I have the solution to this problem, but its understanding some multiplication that's getting me.

They get x=5/4 and y = -3/4

and they do the following

T(c) = xT(a) +yT(b)

T(²₁)=(5/4)T(¹₂)-(3/4)T(^-1₂)

(5/4)(²₃)-(3/4)(²_-3)

(¹₆)

I just need someone to explain to me how they got the 1 and 6 at the end.

5/4 * 2 - 3/4 * 2 = 2/4 * 2 = 1
and
5/4 * 3 - 3/4 * (-3) = 15/4 + 9/4 = 24/4 = 6

All of the expressions are two-d vectors. They are just using ordinary vector arithmetic to get their answer.

BTW, you should connect equal expressions with '='.

 

[Fredrik]

T(xa+yb) = xT(a)+yT(b)

This is a nitpick, but it's far more common to denote the vectors by x,y and the scalars by a,b. (It's not wrong to use your notation, but it could cause confusion).

Also note that the equation is just a part of the statement. The full statement goes like this: For all $a,b\in\mathbb R$ and all $x,y\in\mathbb R^2$, we have $T(ax+by)=aT(x)+bT(y)$.

Let $x,y\in\mathbb R^2$ and $a\in\mathbb R$ be arbitrary. Do you know how $ax$ and $x+y$ are defined? Those definitions are the only things that Mark44 used to answer your question.

 

[HallsofIvy]

Any linear transformation from R² to R² maps (x, y) to (ax+ by, cx+ dy) for some numbers a, b, c, and d. You are told that this linear transformation "maps the point (1,2) to (2,3)" so (a(1)+ b(2), (c(1)+ d(2))= (2, 3) which gives the two equations a+ 2b= 2 and c+ 2d= 3. You are told that this linear transformation also "maps the point (-1,2) to (2,-3)" so -a+ 2b= 2 and -c+ 2d= -3.

Solve the four equations, a+ 2b= 2, c+ 2d= 3, -a+ 2b= 2, and -c+ 2d= -3 for a, b, c, and d.