Engineering Mathematics-I Question Bank Unit-1

Vector Spaces and Scalar Multiplication

• A vector space must satisfy properties under addition and scalar multiplication.
• Scalar multiplication examples tested:
  • (a(x, y, z) = (ax, y, az)): Non-vector space.
  • (a(x, y, z) = (ax, 0, az)): Non-vector space.
  • (a(x, y, z) = (0, 0, 0)): Not a valid scalar multiplication.
  • (a(x, y, z) = (2ax, 2ay, 2az)): Valid vector space.

Sets as Vector Spaces

• Nonnegative real numbers with standard operations fail as a vector space due to lack of negatives.
• Polynomials of degree ≥ 3 lack a zero vector, making it non-vector space.
• Polynomials of degree ≤ 3 form a vector space due to closure under addition and scalar multiplication.
• The set {1, x, x², ...} is not a vector space; not closed under addition.
• Certain matrix forms and conditions (like (2 \times 2) matrices) also tested for being vector spaces based on their properties and operations.

Special Cases of Vector Spaces

• Positive real numbers as a vector space with multiplication for addition must prove closure and properties.
• Ordered pairs (x, y) with modified addition and scalar multiplication form a vector space defined by unique operations.

Subspaces of Polynomial Spaces

• Sets defined by specific conditions (e.g., (f(2) = 1)) do not form subspaces due to lack of closure.
• Structuring through (g(x)) in relation to polynomial degrees checks subspaces of (P3).
• Zero constant terms in polynomials define a valid subspace.

Subspaces of Matrix Spaces

• Matrices with certain properties (e.g., symmetry, specific determinants) were tested for forming valid subspaces.
• Conditions like (A^T = A) show valid subspaces; others based on non-invertibility or specific matrix equations must validate.

Subspaces of Function Spaces

• Functions evaluated on conditions like (f(0) = 0) form a vector space (zero vector condition).
• Maintaining conditions in integration or equality across intervals can also form valid subspaces.

Linear Combinations

• Expressing polynomials as combinations of given forms (e.g. (x + 1), (x^2 + x)) allows for analysis of linear dependencies.
• Specific polynomials demonstrate how they can be represented as linear combinations.

Spanning Sets

• Demonstrating spanning characteristics through basis sets, like vectors in (R^3) or polynomials in (P2), confirms spanned nature.
• Specific examples illustrate how defined sets can reach every vector in a space.

Analysis of Span

• Testing whether elements belong to span confirms closeness within spaces formed by given vectors or matrices.
• Conditions and representations of matrices and concepts play a key role in understanding span properties.