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

In summary, the linear transformation T maps the point (2,1) to (2,3), the point (-1,2) to (2,-3), and the point (5/4, -3/4) to (16).
  • #1
Fellowroot
92
0

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.
 
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  • #2
Fellowroot said:

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 '='.
 
  • #3
Fellowroot said:
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.
 
  • #4
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.
 

1. What is a linear transformation map?

A linear transformation map is a mathematical function that maps a vector in one space to a vector in another space while preserving the linear structure of the original vector. It is commonly used in fields such as mathematics, physics, and engineering to model real-world phenomena and solve complex problems.

2. What are the key properties of a linear transformation map?

The key properties of a linear transformation map are linearity, preservation of vector addition, and preservation of scalar multiplication. This means that the map must satisfy the following equations:
- f(x + y) = f(x) + f(y)
- f(cx) = cf(x)
where x and y are vectors in the original space and c is a scalar.

3. How is a linear transformation map represented mathematically?

A linear transformation map can be represented mathematically using a matrix. The matrix representation of a linear transformation map is obtained by applying the map to each of the standard basis vectors in the original space and writing the resulting vectors as columns in a matrix. This matrix is then used to transform any vector in the original space into its corresponding vector in the new space.

4. What are some common applications of linear transformation maps?

Linear transformation maps have a wide range of applications in various fields. In mathematics, they are used in linear algebra, differential equations, and geometry. In physics, they are used to model systems and phenomena such as motion, fluid dynamics, and electromagnetism. In engineering, they are used in signal processing, control systems, and image processing. They are also used in computer graphics and data analysis.

5. How can one determine if a map is a linear transformation?

To determine if a map is a linear transformation, one can check if it satisfies the key properties of linearity, preservation of vector addition, and preservation of scalar multiplication. If the map satisfies these properties, it is a linear transformation. Additionally, one can also represent the map using a matrix and check if it follows the rules of matrix multiplication and vector transformation. If it does, then it is a valid linear transformation map.

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