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?
Show answer
Matrix multiplication- Why are homogeneous coordinates used in 3D graphics instead of Cartesian coordinates?
Show answer
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?
Show answer
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?
Show answer
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?
Show answer
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?
Show answer
Through interpolation using barycentric coordinates across triangle faces- What linear algebra concept represents the direction perpendicular to a surface in 3D graphics?
Show answer
The normal vector- In vertex shaders, why is linear algebra essential for transforming vertices?
Show answer
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?
Show answer
Multiplication by a rotation matrix, often constructed using Rodrigues' rotation formula- How does the determinant of a transformation matrix relate to its geometric effect?
Show answer
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?
Show answer
Solving the normal equation: w = (X^T X)^-1 X^T y- What role do eigenvalues and eigenvectors play in Principal Component Analysis (PCA)?
Show answer
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?
Show answer
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?
Show answer
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?
Show answer
The number of linearly independent features; full rank means all features contain unique information
FlashKeepers