Is phi(C(u,v))=C(phi(u,v,)) a linear transformation?

In summary, the conversation is discussing whether or not phi(u,v)=(u-2v,-v) is a linear transformation from R^2 to R^2. The first rule for linear transformations states that phi(u1,v1)+(u2,v2)=phi(u1,v1)+phi(u2,v2) must be true. The second rule is phi(C(u,v))=C(phi(u,v)). The person is unsure how to use the second rule and is seeking advice on how to prove or disprove the identity. The suggestion is to try substituting specific values and using the rules for scalar multiplication and the definition of phi to rewrite the equation and better understand it.
  • #1
cad2blender
6
0
Let phi(u,v)=(u-2v,-v) is this a R^2->R^2 a linear transformation?

I know that there must be two rules that must be met in order to be a linear transformation, after doing the first part, it seems that it may be linear. But I do not know how to show whether or not the second rule is satisfied. Any tips?
 
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  • #2
cad2blender said:
Let phi(u,v)=(u-2v,-v) is this a R^2->R^2 a linear transformation?

I know that there must be two rules ... But I do not know how to show whether or not the second rule is satisfied. Any tips?
What is the second rule? Have you plugged this problem and the definition of phi into the rule to see what the result would look like? What problems have you had proving it? How far did you get?
 
  • #3
the thing is that I don't know what to do to use the second rule. The first rule says that phi(u1,v1)+(u2,v2)=phi(u1,v1)+phi(u2,v2) must be true, so far I think i got this to work, after some time I got the right side equal to the left side. That is how far I got.

EDIT: 2nd rule is phi(C(u,v))=C(phi(u,v,))
 
  • #4
cad2blender said:
the thing is that I don't know what to do to use the second rule.

EDIT: 2nd rule is phi(C(u,v))=C(phi(u,v,))
Substituting what you know is always a good thing to try.

You have this equation you want to prove/disprove is an identity. (i.e. it's true for all values of C and (u,v))

Have you tried plugging in some specific values yet? You might get lucky and find a disproof quickly. Always a good thing to try when considering disproving an identity.

You already know two rules that allow you to rewrite parts of this equation. (in particular, the rule for scalar multiplication of vectors, and the definition of phi)

Applying these rules to rewrite the equation you are studying may or may not turn it into something you understand better. But you won't know until you try it.
 

What is a linear transformation?

A linear transformation is a mathematical function that maps a vector in one space to a vector in another space. It preserves the linear structure of the original vector, meaning that the sum of two vectors in the original space will be transformed to the sum of their corresponding transformed vectors in the new space.

What are the properties of a linear transformation?

A linear transformation has two main properties: it preserves addition (meaning that the transformed sum of two vectors is equal to the sum of their transformed individual vectors) and it preserves scalar multiplication (meaning that the transformed vector multiplied by a scalar is equal to the same scalar multiplied by the transformed vector).

How do you represent a linear transformation?

A linear transformation can be represented by a matrix. The columns of the matrix correspond to the transformed basis vectors of the original space, and the rows represent the coordinates of the transformed vector in the new space. The transformation can also be represented by a set of linear equations or by a graph.

What are some common examples of linear transformations?

Some common examples of linear transformations include rotations, reflections, and scaling. These transformations can be represented by specific matrices, and they are commonly used in computer graphics and image processing.

How are linear transformations used in real life?

Linear transformations have many practical applications in various fields, such as physics, engineering, economics, and biology. For example, in physics, linear transformations are used to model the motion of objects and describe the relationship between different physical quantities. In economics, linear transformations are used to analyze supply and demand curves and predict market trends. In biology, linear transformations are used to study genetic inheritance patterns.

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