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The Timpson Lyceum · Linear Algebra (2026)

2.8 Exercises

Problem sheet · 20 points · 4 extra credit

Subspaces of Rn\mathbb{R}^n

Problems 2–4 and 7 use the matrix

A=[132426571315].A = \begin{bmatrix} 1 & 3 & 2 & 4 \\ 2 & 6 & 5 & 7 \\ 1 & 3 & 1 & 5 \end{bmatrix}.

  1. 1. (3 pts) Determine whether each set is a subspace of R3\mathbb{R}^3. Justify each answer.

    (a) H1={(x,y,z):x+2y−z=0}H_1 = \{(x, y, z) : x + 2y - z = 0\}.

    (b) H2={(x,y,z):x≥0}H_2 = \{(x, y, z) : x \ge 0\}.

    (c) H3={(x,y,z):x+y+z=1}H_3 = \{(x, y, z) : x + y + z = 1\}.

  2. 2. (3 pts) Find a basis for Col⁡A\operatorname{Col} A.

  3. 3. (3 pts) Find a basis for Nul⁡A\operatorname{Nul} A.

  4. 4. (2 pts) Determine whether b=[372]\mathbf{b} = \begin{bmatrix} 3 \\ 7 \\ 2 \end{bmatrix} is in Col⁡A\operatorname{Col} A.

  5. 5. (3 pts) State whether each is true or false, and justify. Let AA be an m×nm\times n matrix.

    (a) Nul⁡A\operatorname{Nul} A is a subspace of Rm\mathbb{R}^m.

    (b) Col⁡A\operatorname{Col} A is a subspace of Rm\mathbb{R}^m.

    (c) A single nonzero vector v\mathbf{v} in Rn\mathbb{R}^n spans a subspace (the line through the origin in the direction of v\mathbf{v}).

  6. 6. (3 pts) Prove directly from the definition of a subspace that Nul⁡A\operatorname{Nul} A is a subspace of Rn\mathbb{R}^n for any m×nm\times n matrix AA.

  7. 7. (3 pts) Determine whether v=[111]\mathbf{v} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} is in Col⁡A\operatorname{Col} A, and explain what your answer says about the solvability of Ax=vA\mathbf{x} = \mathbf{v}.

  8. 8. (4 pts ••) Let UU and WW be subspaces of Rn\mathbb{R}^n. Prove that U∩WU \cap W is a subspace of Rn\mathbb{R}^n, and give an explicit example showing that U∪WU \cup W need not be a subspace.