Let R^n be the set of real numbers, U be a subspace of R^n, and M be the matrix corresponding to the projection onto the subspace U. Which of the following statements is/are TRUE? A. If U is a 1-dimensional subspace of R^n, then the null space of M is a 1-dimensional subspace. B. If U is a 2-dimensional subspace of R^n, then the null space of M is a (n-2)-dimensional subspace. C. M^2 = M. D. M^T = M.

GATE 2024 · Engineering Mathematics · Vector Space · medium

Answer: B, C, D

  1. Option C: M^2 = M (idempotence): For any x in R^n, Mx lies in U. Projecting a vector already in U onto U leaves it unchanged, so M(Mx) = Mx. Therefore M^2 = M. Option C is TRUE.
  2. Option D: M^T = M (symmetry): The orthogonal projection matrix onto U is given by M = X(X^T X)^(-1) X^T (when X is a matrix whose columns span U). This is visibly symmetric. Equivalently, the orthogonality condition (x - Mx) perpendicular to U for all x forces M^T = M. Option D is TRUE.
  3. Option B: Null space of M when dim(U)=2: rank(M) = dim(U) = 2 (M maps onto U). By the rank-nullity theorem: dim(null(M)) = n - 2. So the null space is (n-2)-dimensional. Option B is TRUE.
  4. Option A: Null space of M when dim(U)=1: When dim(U)=1, dim(null(M)) = n - 1. Option A claims the null space is 1-dimensional. This is FALSE unless n=2. For general n >= 2, the null space has dimension n-1, not 1.