PCA using SVD Practice Quiz for Linear Algebra.

📚 Topic Summary

Principal Component Analysis (PCA) using Singular Value Decomposition (SVD) is a powerful technique for dimensionality reduction. It identifies the principal components of your data, which are the directions of maximum variance. SVD helps to decompose the data matrix into orthogonal components, making it easier to extract these principal components. Essentially, you're finding a new set of uncorrelated variables that explain the most variance in your original dataset, which is super useful for simplifying complex data and improving machine learning model performance. Think of it like finding the most important features in a dataset!

🧠 Part A: Vocabulary

Match the term to its correct definition:

1. Term: Singular Value
2. Term: Principal Component
3. Term: Eigenvector
4. Term: Variance
5. Term: Dimensionality Reduction

1. Definition: A vector that does not change direction when a linear transformation is applied to it.
2. Definition: A measure of how spread out the data is.
3. Definition: A technique to reduce the number of variables in a dataset.
4. Definition: A measure of the 'strength' of each principal component.
5. Definition: A new variable that is a linear combination of the original variables and captures the most variance.

✏️ Part B: Fill in the Blanks

PCA uses SVD to decompose a data matrix into three matrices: $U$, $\Sigma$, and $V^T$. The matrix $U$ contains the __________, the matrix $\Sigma$ contains the __________, and the matrix $V$ contains the __________. The principal components are derived from the columns of the __________ matrix.

🤔 Part C: Critical Thinking

Explain, in your own words, why PCA using SVD is useful in real-world applications, providing at least two concrete examples where it is beneficial.