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SOLVED: 2.2.24. Let W be a subspace of V. A subspace Z in V is called a complementary subspace to W if (i) W ∩ Z = 0, and (ii) W Z = V, i.e., every v in V can be written as v = w z for w in W and z in Z. Show that the x- and y-axes are complementary subspaces of R^2. (b) Show that the lines y and r = 3y are complementary subspaces of R^2. Show that the line a, 2a, 3a and the plane 2y 3z = 0 are complementary subspaces of R^3. (d) Prove that if v = w 2, then w in W and z in Z are unique
SOLVED: 2.2.24. Let W be a subspace of V. A subspace Z in V …
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SOLVED: Let T : V -> W be a linear transformation between vectorspaces. Prove: (i) Ker T = x in V | T(x) = 0 is a subspace of V.[Use subspace test.](ii) Ran T is a subspace of W. [You can again use subspacetest.]
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