Lesson 4 · Examples and problems

Practice: The four subspaces

Three worked examples, then five problems. rref(A) and np.linalg.matrix_rank(A) in the lab will check your pivots and ranks.

Worked examples

Is the set of all (x, y, z) with x = 2y a subspace of 3D space?

Zero: (0, 0, 0) has 0 = 2·0 ✓

Add: if x₁ = 2y₁ and x₂ = 2y₂, then x₁ + x₂ = 2(y₁ + y₂) ✓

Scale: cx₁ = 2(cy₁) ✓

Yes. It is a plane through the origin. With x = 2y + 1 it would fail at once: the origin is missing.

Find the nullspace of A = 123246.

Row 2 − 2·row 1 = 0, so R = 123000. One pivot (column 1), rank 1.

Free: x₂, x₃. x₂ = 1, x₃ = 0 gives (−2, 1, 0). x₂ = 0, x₃ = 1 gives (−3, 0, 1).

N(A) is all combinations c·(−2, 1, 0) + d·(−3, 0, 1): a plane, dimension n − r = 3 − 1 = 2.

Solve x + 2y + 3z = 6.

Particular solution: set the free variables y = z = 0, then x = 6. x_p = (6, 0, 0)

Nullspace: the same matrix as Example 2's first row, special solutions (−2, 1, 0) and (−3, 0, 1)

x = (6, 0, 0) + c·(−2, 1, 0) + d·(−3, 0, 1)

Geometrically: a plane parallel to the nullspace, shifted to pass through (6, 0, 0).

Which of these are subspaces of the plane? (a) all (x, y) with x = y (b) all (x, y) with x + y = 1 (c) all (x, y) with xy = 0.

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(a) Yes: a line through the origin.

(b) No: (0, 0) is not on it.

Find the nullspace of 1−12−2.

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Both rows say x − y = 0. So N(A) is every multiple of (1, 1): a line.

Check: A(1, 1) = (1 − 1, 2 − 2) = (0, 0) ✓

Find R, the rank and a special solution for A = 121243364.

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Row 2 − 2·row 1 = (0, 0, 1), row 3 − 3·row 1 = (0, 0, 1), then row 3 − row 2 = 0. Clearing above the pivot: R = 120001000.

Pivots in columns 1 and 3, so rank 2. Column 2 is free: x₂ = 1 gives the special solution (−2, 1, 0).

Are (1, 2, 0), (0, 1, 1) and (1, 3, 1) independent?

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No. (1, 3, 1) = (1, 2, 0) + (0, 1, 1), so 1·v₁ + 1·v₂ − 1·v₃ = 0. The rank is 2 and the three vectors lie in one plane.

A is 3 by 5 with rank 2. Give the dimension of each of the four subspaces.

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m = 3, n = 5, r = 2.

Row space r = 2, nullspace n − r = 3, column space r = 2, left nullspace m − r = 1.