Let U and W be subspaces of the vector space V. The intersection UW is also a subspace of V, whereas the union UW is, in general, not a subspace. Define the sum of the subspaces U and W as follows. U + W = (u + w : U epsilon U, w epsilon W) 1. Prove that U + W is a subspace of V. 2. Consider the subspaces of V = R^3 defined below. U= {(x, y, x - y): x, y epsilon R) W = ((X, 0, x): x epsilon R) Z = {(x,x,x):x,epsilon R) Find +W, U + Z, and W + Z. 3. If U and W are subspaces of V such that V = U + W and U W = (0), prove that every vector in V has a unique representation of the form u + w, where u is in U and w is in W. In this case, we say that V is the direct sum of U and W and write V = U W. Which of the sums in part 2 are direct sums? 4. Let V = U W and suppose that {u1,u2, u3,... Uk} is a basis for the subspace U and {w1,w2, w3,... Wn} is a basis for W. Prove that the set {u1,u2, u3,... uk, w1 w2, w3,... wn) is a basis for V. 5. Consider the subspaces of V = R^3 defined below. U = {(x, 0, y): x, y epsilon R} W = {(0, y): x, y epsilon R} Show that R^3 = U + W. Is R3 the direct sum of U and W? What are the dimensions of U, W, U (1 W and U + W? In general, formulate a conjecture that relates the dimensions of U, W, U f1 W and U + W
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Let U and W be subspaces of the vector space V. The intersection UW is also a subspace of V, whereas the union UW is, in general, not a subspace. Define the sum of the subspaces U and W as follows. U + W = (u + w : U epsilon U, w epsilon W) 1. Prove that U + W is a subspace of V. 2. Consider the subspaces of V = R^3 defined below. U= {(x, y, x - y): x, y epsilon R) W = ((X, 0, x): x epsilon R) Z = {(x,x,x):x,epsilon R) Find +W, U + Z, and W + Z. 3. If U and W are subspaces of V such that V = U + W and U W = (0), prove that every vector in V has a unique representation of the form u + w, where u is in U and w is in W. In this case, we say that V is the direct sum of U and W and write V = U W. Which of the sums in part 2 are direct sums? 4. Let V = U W and suppose that {u1,u2, u3,... Uk} is a basis for the subspace U and {w1,w2, w3,... Wn} is a basis for W. Prove that the set {u1,u2, u3,... uk, w1 w2, w3,... wn) is a basis for V. 5. Consider the subspaces of V = R^3 defined below. U = {(x, 0, y): x, y epsilon R} W = {(0, y): x, y epsilon R} Show that R^3 = U + W. Is R3 the direct sum of U and W? What are the dimensions of U, W, U (1 W and U + W? In general, formulate a conjecture that relates the dimensions of U, W, U f1 W and U + W