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2. If U and W are subspaces of V such that V = U+W and UnW = {0}, then 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. V is called the direct sum of U and W, and is written as V=U @W. Direct sum

2. If U and W are subspaces of V such that V = U+W and UnW = {0}, then 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. V is called the direct sum of U and W, and is written as V=U @W. Direct sum

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. If U and W are subspaces of V such that V = U+W and UnW = {0}, then 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. V is called the direct sum of U and W, and is written as V=U @W. Direct sum Which of the sums in part (1) are direct sums? 3. Let V=UW, and let {u₁, 2,..., uk} be a basis for the subspace U and {w₁, W2,..., Wm} be a basis for the subspace W. Prove that the set {u₁,..., uk, w₁,..., Wm} is a basis for V. 4. Consider the subspaces U = {(x, 0, y): x, y ≤ R} and W = {(0, x, y): x, y = R} of V = R³. Show that
Transcribed Image Text:2. If U and W are subspaces of V such that V = U+W and UnW = {0}, then 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. V is called the direct sum of U and W, and is written as V=U @W. Direct sum Which of the sums in part (1) are direct sums? 3. Let V=UW, and let {u₁, 2,..., uk} be a basis for the subspace U and {w₁, W2,..., Wm} be a basis for the subspace W. Prove that the set {u₁,..., uk, w₁,..., Wm} is a basis for V. 4. Consider the subspaces U = {(x, 0, y): x, y ≤ R} and W = {(0, x, y): x, y = R} of V = R³. Show that
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