Linear Transformations Rn->Rm Question

In summary, the question involves determining if the given transformation T from Rn to Rm is linear, and the solution manual provides a proof using the theorem that a transformation is linear if and only if certain relationships hold for all vectors and scalars. The proof involves setting two vectors u and v and showing that T(u+v) is equal to T(u) + T(v). The manual also includes the proof using the second condition of the theorem, which was accidentally left out in the original post.
  • #1
haribol
52
0
Linear Transformations Rn-->Rm Question

I would be very grateful if someone can explain what is going on in the following problem:

Determine whether the following T:Rn to Rm

T(x,y)=(2x,y)

Solution from solutions manual:

T((x1,y1) + (x2,y2)) = (2(x1+x2), y1+y2) = (2x1,y1) + (2x2,y2) = T(x1,y1) + T(x2,y2)

My questions are

1. Where did the x1's and the x2's and the y1's and the y2's come from?

2. Can you please explain what's happening step by step?



[PS]--> The questions asks to use the theorem which states:

A transformation T:Rn --> Rm is linear if and only if the following relationships hold for all vectors u and v in Rn and every scalar c

a) T(u+v) = T(u) + T(v)

b)T(cu) = cT(u)
 
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  • #2
haribol said:
I would be very grateful if someone can explain what is going on in the following problem:

Determine whether the following T:Rn to Rm

T(x,y)=(2x,y)

Solution from solutions manual:

T((x1,y1) + (x2,y2)) = (2(x1+x2), y1+y2) = (2x1,y1) + (2x2,y2) = T(x1,y1) + T(x2,y2)

My questions are

1. Where did the x1's and the x2's and the y1's and the y2's come from?

2. Can you please explain what's happening step by step?



[PS]--> The questions asks to use the theorem which states:

A transformation T:Rn --> Rm is linear if and only if the following relationships hold for all vectors u and v in Rn and every scalar c

a) T(u+v) = T(u) + T(v)

b)T(cu) = cT(u)

He has set [itex]\vec{u} = (x_1,y_1), \ \ \vec{v} = (x_2,y_2)[/itex] and showed using vector addition properties that [itex]T(\vec{u}+\vec{v}) = T(\vec{u})+T(\vec{v})[/itex]
This proof is imcomplete though because he left out condition b).
 
  • #3
Thank you quasar987 for the clarification. The manual does include the proof using condition b) but I forgot to type it.

Thanks for that clarification.
 

1. What is a linear transformation from Rn to Rm?

A linear transformation from Rn to Rm is a mathematical function that maps vectors from n-dimensional space to m-dimensional space while preserving the properties of linearity. This means that the transformation preserves operations such as addition and scalar multiplication.

2. How is a linear transformation represented mathematically?

A linear transformation can be represented mathematically using a matrix. The matrix has m rows and n columns, where m is the dimension of the output space and n is the dimension of the input space. The entries of the matrix represent the coefficients of the linear combination of the input vectors that make up the output vectors.

3. What are the properties of a linear transformation?

A linear transformation has three main properties: it preserves vector addition, it preserves scalar multiplication, and it preserves the zero vector. This means that the transformation of the sum of two vectors is equal to the sum of the individual transformations and the transformation of a scalar multiple of a vector is equal to the scalar multiple of the transformation of the vector. Additionally, the transformation of the zero vector is always the zero vector.

4. How is a linear transformation different from other types of transformations?

A linear transformation is different from other types of transformations, such as nonlinear or affine transformations, because it preserves linearity. This means that the transformation of a line is always a line, and the transformation of a plane is always a plane. Nonlinear and affine transformations do not have this property and can distort or change the shape of the original object.

5. How are linear transformations used in real life?

Linear transformations have many real-life applications, such as in computer graphics, data analysis, and engineering. In computer graphics, linear transformations are used to rotate, scale, and translate objects on the screen. In data analysis, linear transformations are used to reduce the dimensionality of data and visualize high-dimensional data. In engineering, linear transformations are used to model systems and make predictions based on input parameters.

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