How linear algebra concepts are applied in computer graphics, machine learning, and data science, one application per card.
29 cards · basic cards · AI-written, checked twice. Edit anything.
- In 3D computer graphics, what linear algebra operation combines multiple transformations (rotation, scaling, translation) into a single operation?
- Matrix multiplication
- Why are homogeneous coordinates used in 3D graphics instead of Cartesian coordinates?
- They allow affine transformations, including translation, to be represented as matrix multiplication
- What matrix operation is used to compute perspective projection from 3D world coordinates to 2D screen coordinates?
- Multiplying by the projection matrix, then dividing by the homogeneous coordinate
- In graphics lighting, how does the dot product of a surface normal vector with a light direction vector determine illumination?
- The dot product indicates the angle between them; higher values mean more direct light and brighter illumination
- What is the role of the view matrix in 3D graphics?
- It transforms world coordinates into camera (eye) coordinates, positioning and orienting the camera in the scene
- How are texture coordinates mapped to 3D model surfaces using linear algebra?
- Through interpolation using barycentric coordinates across triangle faces
- What linear algebra concept represents the direction perpendicular to a surface in 3D graphics?
- The normal vector
- In vertex shaders, why is linear algebra essential for transforming vertices?
- Vertices must be transformed through multiple coordinate systems (object, world, view, clip) using matrix multiplication
- What matrix operation is used to rotate a 3D point around an arbitrary axis?
- Multiplication by a rotation matrix, often constructed using Rodrigues' rotation formula
- How does the determinant of a transformation matrix relate to its geometric effect?
- Its absolute value represents how the transformation scales volume; its sign indicates if orientation is flipped
- In linear regression, what linear algebra operation finds the optimal weight vector minimizing prediction error?
- Solving the normal equation: w = (X^T X)^-1 X^T y
- What role do eigenvalues and eigenvectors play in Principal Component Analysis (PCA)?
- Eigenvectors of the covariance matrix become the principal components; eigenvalues represent variance explained by each component
- How does gradient descent use vectors and matrices in neural networks?
- It computes the gradient vector of the loss function and updates weights by stepping opposite to the gradient direction
- What is the purpose of Singular Value Decomposition (SVD) in machine learning?
- It decomposes a matrix into U, Sigma, and V^T, useful for dimensionality reduction, noise filtering, and data analysis
- In machine learning, what does the rank of a feature matrix indicate?
- The number of linearly independent features; full rank means all features contain unique information
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