Linear Algebra and Calculus DY0-001 Practice Questions (Page 4)

Questions 16–20

1. 16application · medium

   A data engineer is implementing a linear transformation in a data pipeline. They have a matrix A = [[1, 2], [3, 4]] and a vector v = [5, 6]. They need to compute the product A * v to transform the vector. What is the resulting vector?

   • A[11, 22]
   • B[23, 34]
   • C[17, 39]
   • D[32, 50]
   Reveal Solution Discussion Select an answer first
2. 17foundation · easy

   What does the rank of a matrix represent?

   • AThe number of rows in the matrix
   • BThe number of linearly independent rows or columns
   • CThe total number of nonzero entries in the matrix
   • DThe determinant of the matrix
   Reveal Solution Discussion Select an answer first
4. 18application · medium

   A machine learning engineer is implementing a linear regression model using the normal equation. They have a design matrix X of shape (1000, 50) and a target vector y of shape (1000, 1). To compute the optimal weights, they need to calculate (X^T X)^(-1) X^T y. However, they suspect that some features are linearly dependent. What is the most appropriate action to take before computing the inverse?

   • ACheck the rank of X^T X; if it is less than 50, use regularization or remove dependent features.
   • BTranspose X before computing the inverse to avoid singularity.
   • CCompute the inverse directly; the normal equation always works regardless of feature dependence.
   • DIncrease the number of samples to 2000 to ensure the matrix is invertible.
   Reveal Solution Discussion Select an answer first
5. 19foundation · easy

   For the function f(x, y) = x^2 + 3xy, what is the partial derivative with respect to x?

   • A2x + 3
   • B2x + 3x
   • C2x + 3y
   • D2x + 3y + 3x
   Reveal Solution Discussion Select an answer first
6. 20foundation · easy

   Which condition must be true for two matrices to be multiplied?

   • AThe number of columns in the first matrix must equal the number of rows in the second matrix.
   • BBoth matrices must have the same number of rows and columns.
   • CThe number of rows in the first matrix must equal the number of columns in the second matrix.
   • DBoth matrices must be square and have the same size.
   Reveal Solution Discussion Select an answer first