Linear Transformations Problem

In summary, the conversation is about finding the standard matrix for a linear transformation with the transformation rule T(v) = kv for v in R^n. The person asking for help is advised to pick a usual basis in R^n and evaluate T on the basis vectors to get the column vectors of the matrix.
  • #1
RandR
2
0
Hello,
Can someone help me with this problem? Thanks in advance
Let T be a linear transformation such that T (v) = kv for v in R^n.
Find the standard matrix for T.
 
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  • #2
Linear Transformations Problem - Please Help

Hello,
Can someone help me with this problem? Thanks in advance
Let T be a linear transformation such that T (v) = kv for v in R^n.
Find the standard matrix for T.
 
  • #3
Can you show your attempt?
Do you know what an eigenvector is?
 
  • #4
Pick the usual basis in R^n, evaluate T on those basis vectors. The results are the column vectors of the matrix of T. What does it look like?
 
  • #5
Cute (and almost trivial) problem!
 

1. What is a linear transformation?

A linear transformation is a function that maps one vector space to another while preserving the basic structure of the vector space. This means that the transformation must preserve addition and scalar multiplication.

2. How do you represent a linear transformation?

A linear transformation can be represented using a matrix. The columns of the matrix represent the images of the basis vectors of the domain, while the rows represent the coordinates of the images in the range.

3. What is the difference between a linear transformation and a non-linear transformation?

A linear transformation preserves the basic structure of a vector space, while a non-linear transformation does not. This means that a non-linear transformation does not preserve addition and scalar multiplication.

4. How do you determine if a transformation is linear?

To determine if a transformation is linear, you can check if it preserves addition and scalar multiplication. This can be done by applying the transformation to the sum of two vectors and comparing it to the sum of the transformed vectors, as well as applying the transformation to a vector multiplied by a scalar and comparing it to the transformed vector multiplied by the same scalar.

5. What are some real-life applications of linear transformations?

Linear transformations have various applications in fields such as engineering, computer graphics, and economics. They can be used to rotate, scale, and skew images in computer graphics, model systems in engineering, and analyze demand and supply in economics.

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