Let ℝ be the set of real numbers, π‘ˆ be a subspace of ℝ³ and 𝑴 ∈ ℝ³×³ be the…

2024

Let ℝ be the set of real numbers, π‘ˆ be a subspace of ℝ³ and 𝑴 ∈ ℝ³×³ be the matrix corresponding to the projection onto the subspace π‘ˆ. Which of the following statements is/are TRUE?

  1. A.

    If π‘ˆ is a 1-dimensional subspace of ℝ³, then the null space of 𝑴 is a 1-dimensional subspace.

  2. B.

    If π‘ˆ is a 2-dimensional subspace of ℝ³, then the null space of 𝑴 is a 1-dimensional subspace.

  3. C.

    𝑴² = 𝑴

  4. D.

    𝑴³ = 𝑴

Show answer & explanation

Correct answer: B, C, D

Key facts for the projection matrix 𝑴 onto a subspace π‘ˆ of ℝ³: Range(𝑴) = π‘ˆ, so rank(𝑴) = dim(π‘ˆ). By the rank-nullity theorem in ℝ³, nullity(𝑴) = 3 - dim(π‘ˆ). Projection matrices are idempotent, so 𝑴² = 𝑴. Therefore 𝑴³ = 𝑴²𝑴 = 𝑴𝑴 = 𝑴² = 𝑴. Statement 1 is false because if dim(π‘ˆ) = 1, then nullity(𝑴) = 2, not 1. Statement 2 is true because if dim(π‘ˆ) = 2, then nullity(𝑴) = 1. Statements 3 and 4 are true because 𝑴² = 𝑴 and hence 𝑴³ = 𝑴. Therefore, statements 2, 3, and 4 are TRUE.

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