MATHEMATICAL FOUNDATIONS FOR DATA SCIENCE End Semester Exam Questions

Vector Space and Subspace

• A non-empty subset W of a vector space V is a subspace of V if it satisfies the following condition: W is closed under vector addition and scalar multiplication.
• In other words, for all u, v in W and a in ℝ, u + v is in W and au is in W.

Eigenvalues and Eigenvectors

• The eigenvalues of the matrix A = | 2 3 \ 1 4 | are λ₁ = 5 and λ₂ = 1.
• The eigenvectors corresponding to λ₁ = 5 are vectors of the form (3t, t), where t is a non-zero real number.
• The eigenvectors corresponding to λ₂ = 1 are vectors of the form (t, -t), where t is a non-zero real number.

Gram-Schmidt Process

• Applying the Gram-Schmidt process to P₁ = (1,0,1), P₂ = (1,0,1), P₃ = (0,3,4) yields an orthonormal set of vectors:
  • Q₁ = (1/√2, 0, 1/√2)
  • Q₂ = (0, 0, 0) (since P₂ is a linear combination of P₁)
  • Q₃ = (3/√10, 1/√10, 2/√10)

Matrix Operations

• The rank of a matrix is the maximum number of linearly independent rows or columns.
• The echelon form of a matrix is a row-equivalent matrix with leading entries of 1, and all other entries in the same column are 0.
• The normal form (or reduced row echelon form) of a matrix is an echelon form with all entries below the leading entries being 0.

Canonical Form

• The canonical form of the matrix | 2 2 2 \ 4 3 0 \ 3 1 -4 | is:
  • | 1 0 0 \ 0 1 0 \ 0 0 1 |

Linear Transformation

• A transformation T: ℝ³ → ℝ³ is a linear transformation if it satisfies the following conditions:
  • T(u + v) = T(u) + T(v) for all u, v in ℝ³
  • T(au) = aT(u) for all u in ℝ³ and a in ℝ
• The transformation T(x₁, x₂, x₃) = (x₁ - x₂ + 2x₃, 2x₁ + x₂, -x₁ - 2x₂ + 2x₃) is a linear transformation.