How to Show a Function is Not a Linear Transformation

TL;DR

A function in R^2 is shown to be nonlinear by providing an example that breaks the linearity condition.

Transcript

in this problem we have a function from r squared into r squared given by f of x y equals xy comma x and we have to show our function is not linear so recall a function is linear if you have f of a vector v plus a vector w equal to f of v plus f of w and if you have f of a scalar times a vector v that's equal to c times f of the vector and this is ... Read More

Key Insights

• ❓ The linearity condition is an essential criterion to determine if a function is linear or not.
• 👍 Finding just one example that breaks the linearity condition is sufficient to prove that a function is not linear.
• 🈸 Calculations involving vectors and the function's application to them are necessary to compare the results and demonstrate linearity or nonlinearity.
• 🥰 The provided example of v = [2, 3] and w = [3, 4] shows that the function is not linear.

Questions & Answers

Q: What is the definition of a linear function?

A linear function satisfies the condition that the function applied to the sum of two vectors equals the sum of the function applied to each vector separately.

Q: How can we prove a function is not linear?

To prove a function is not linear, we only need to find one example where the linearity condition does not hold true.

Q: What vectors are chosen as an example in this video?

The example chosen in the video includes the vectors v = [2, 3] and w = [3, 4].

Q: How is it shown that the function is not linear with the chosen example?

By calculating the function applied to v + w and comparing it to the sum of the function applied to v and w separately, it is shown that they are not equal.

Summary & Key Takeaways

• The video discusses a given function in R^2 and aims to prove that it is not linear.

• By applying the linearity condition to the function, an example is provided where it fails to hold true.

• This example involves finding vectors, calculating the function applied to them, and comparing the results to disprove linearity.