Very simple linear algebra question:

In summary, The conversation is about understanding the concept of a linear transformation and how it applies to the given equations. The criteria for a transformation to be considered linear is that it must satisfy T(cv) = cT(v) and T(v+w) = T(v) + T(w) for all scalars c and vectors v and w. The specific equation y_2 = x_1*x_3 is being examined and it is noted that x(x_1*x_3) is not equal to x_1*x(x_3). The questioner asks for further explanation on how to proceed.
  • #1
frasifrasi
276
0
This is a very basic question, but I am not understanding the concept here.

it asks, consider the linear transformation:

y_1(y subscript 1) = x_2 - x_3
y_2 = x_1*x_3
y_3 = x_1 - x_2

Can anyone explain why this is not linear? I am not sure what the criteria is and the book doesn't have any example like this reasoned out.

Thank you.
 
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  • #2
A transformation T is linear if T(cv) = cT(v) and T(v+w) = T(v) + T(w) for all scalars c and vectors v and w. Does this help?
 
  • #3
In particular, look at what happens to y2.
 
  • #4
Ok, so for y_2:

x(x_1*x_3) is not the same as...?

I am not sure how to apply this. Can someone explain how to proceed?
 

What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It involves the use of matrices, vectors, and other mathematical structures to solve problems related to systems of linear equations.

What are the basic concepts of linear algebra?

The basic concepts of linear algebra include vectors, matrices, linear transformations, and systems of linear equations. Vectors are objects with both magnitude and direction, while matrices are arrays of numbers used to represent linear transformations. Linear transformations are functions that map vectors from one space to another, and systems of linear equations involve solving equations with multiple variables.

What is a vector space?

A vector space is a mathematical structure that consists of a set of vectors and operations such as addition and scalar multiplication. These operations follow specific rules, such as closure, associativity, and distributivity, and allow for the manipulation and analysis of vectors in a systematic way.

What is the importance of linear algebra in science?

Linear algebra plays a crucial role in many scientific fields, including physics, engineering, computer science, and economics. It allows for the representation and manipulation of complex systems and data, making it an essential tool for solving problems and making predictions in these fields.

How can linear algebra be applied in real life?

Linear algebra has many practical applications in everyday life, such as in computer graphics, data analysis, and optimization problems. It is also used in fields such as finance and economics to model and predict market trends. Additionally, linear algebra is used in engineering for tasks like image and signal processing, as well as in physics for understanding physical systems and their behavior.

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