Change of basis is an important idea in linear algebra. It helps us understand how to change coordinates, which is really useful in many areas. Here are some ways change of basis is used in real life:

1. Computer Graphics:

   • In computer graphics, we need to change how we represent objects and their movements using coordinates.
   • Change of basis is important when designing graphics for different screens or in 3D spaces.
   • For example, when we switch a model from its local coordinates to world coordinates, then to camera coordinates, and finally to screen coordinates, we do a lot of basis changes. About 72% of graphic applications use some type of basis change.

2. Data Science and Machine Learning:

   • In data science, we often use a method called Principal Component Analysis (PCA) for reducing dimensions. This method relies on change of basis.
   • By changing the data into a new set of axes that show the direction of the most variation, we can make complex datasets easier to understand.
   • PCA can take something like 50 dimensions and reduce it to just 2 or 3 while keeping over 95% of the information. This helps with visualizing data and can make processing quicker by up to 80%.

3. Signal Processing:

   • In signal processing, changing basis helps us move signals into different areas, like time or frequency.
   • Techniques like the Fourier Transform use basis changes to analyze signals better.
   • The Fast Fourier Transform (FFT) makes the calculations easier and quicker, changing complexity from $O(N^2)$ to $O(N \log N)$. This is really useful in telecommunications, where more than 90% of signals processed use FFT.

4. Robotics and Control Systems:

   • In robotics, understanding how to change basis helps in controlling robot movements better.
   • We use transformations to connect joint coordinates to regular coordinates to plan paths and control motion.
   • Around 65% of robotic systems in factories use coordinate transformations to work effectively.

5. Quantum Mechanics:

   • In quantum mechanics, the state of a quantum system can be shown in different bases, like position or momentum.
   • Change of basis lets us switch between these forms based on the problem we are working on.
   • In quantum computing, we often look at qubit states in different bases, which is really important for quantum algorithms. Research shows that about 80% of quantum algorithms do better when they use basis changes.

In conclusion, change of basis is a powerful concept in linear algebra and is used in many fields, like computer graphics, data science, signal processing, robotics, and quantum mechanics. Each area uses it to handle complex changes, improve performance, and make calculations more efficient. This work greatly impacts technology and research in positive ways.