SUMMARY
The gradient of the function f(X) = C^T * X, where C is a column vector, is definitively equal to C, not C^T. This is due to the definition of the gradient in relation to vector calculus, where the gradient of a scalar function with respect to a vector yields a column vector. The discussion emphasizes the importance of distinguishing between row vectors and column vectors when calculating gradients, as they yield different results based on their orientation.
PREREQUISITES
- Understanding of vector calculus and gradients
- Familiarity with matrix operations, specifically transposition
- Knowledge of scalar functions and their derivatives
- Basic concepts of linear algebra, including vector representation
NEXT STEPS
- Study the properties of gradients in vector calculus
- Learn about the implications of row vs. column vectors in matrix operations
- Explore the derivation of gradients for different types of functions
- Investigate applications of gradients in optimization problems
USEFUL FOR
Mathematicians, data scientists, and anyone involved in machine learning or optimization who needs a clear understanding of gradient calculations in matrix functions.