Q1. Scenario: You are a data scientist analyzing customer data for an e-commerce site. You have 1000 customers and 50 features (age, purchase history, etc.). How would you represent this data in a structured way?
This data is naturally represented as a matrix with 1000 rows (customers) and 50 columns (features). Each entry is the value of a feature for a customer. Matrices allow you to perform operations like scaling, transforming, or applying machine learning algorithms (e.g., linear regression: y = X·w). Libraries like NumPy and pandas store data as matrices for efficient computation.
Q2. Scenario: In image processing, a grayscale image is a matrix of pixel intensities (0-255). If you have a 100x100 image and you want to blur it using a 3x3 averaging kernel, how would you apply this kernel using matrix operations?
Convolution: Slide the kernel over the image matrix. For each position, multiply the kernel element-wise with the overlapping image patch and sum. This yields a new matrix of the same size (with padding). Matrix convolution is the basis of edge detection, blurring, and feature extraction in neural networks (CNNs).
Q3. Scenario: A social network has users as rows and columns, and matrix entry (i,j) = 1 if user i follows user j. How would you find the number of mutual follows between users? What operation gives the number of two-step connections?
The matrix is symmetric for mutual follows (M[i,j] = M[j,i] = 1). Count mutual follows by summing the lower triangle of the matrix. For two-step connections (follows of follows), multiply the matrix by itself: M²[i,j] counts the number of paths of length 2 from i to j. This is a key concept in network analysis (e.g., PageRank algorithm).
Q4. Scenario: In a recommendation system, you have a user-item matrix where entries are ratings. This matrix is very sparse (most entries empty). How would you find similar users without filling all empty cells?
Use matrix factorization: decompose the matrix into two smaller matrices U (user factors) and V (item factors) such that U·V^T approximates the rating matrix. Then user similarity can be computed from U. This is the basis of collaborative filtering algorithms used by Netflix and Amazon.
Q5. Scenario: You have a matrix of pixel values for a batch of 100 grayscale images, each 28x28 pixels. What is the shape of this matrix if you flatten each image into a vector? How would you store a batch for neural network input?
Flatten each 28x28 image into a 1D vector of length 784. The batch matrix becomes 100 rows (images) × 784 columns (pixels). This is the standard input format for fully connected neural networks. Then you can multiply by weight matrices to compute predictions.