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77. x(x - 1) = 0. 78. x² + 2y² + 1 = 0. 79. xyz = 0.

77. x(x - 1) = 0. 78. x² + 2y² + 1 = 0. 79. xyz = 0.

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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Parts 77-80 Please

**Problem IV:** Consider the set of all vectors \((x, y, z) \in \mathbb{R}^3\) satisfying the following conditions. Determine whether it is a subspace of the vector space \(\mathbb{R}^3\) or not. Briefly explain. 61. \(x - y + z = 0\). 62. \(x + y + z = 1\). 63. \(x + y - z = 0\). 64. \(x = y = z = 0\). 65. \(x - y = z + 1 = 0\). 66. \(x - y = y = z - x\). 67. \(x = y = 0\). 68. \(y = z = 1\). 69. \(x + y + z \geq 0\). 70. \(x = y \text{ and } z \geq 0\). 71. \(xy \geq 0\). 72. \(|x| = |y|\). 73. \(xy \geq 0 \text{ and } |x| = |y|\). 74. \(z = x^2 - y^2\). 75. \(z = x^2 + y^2\). 76. \(x^2 + y^2 + z^2 = 1\). 77. \(x(x - 1) = 0\). 78. \(x^2 + 2y^2 + 1 = 0\). 79. \(xy = z = 0\). 80. \(xy = y = 0\).
Transcribed Image Text:**Problem IV:** Consider the set of all vectors \((x, y, z) \in \mathbb{R}^3\) satisfying the following conditions. Determine whether it is a subspace of the vector space \(\mathbb{R}^3\) or not. Briefly explain. 61. \(x - y + z = 0\). 62. \(x + y + z = 1\). 63. \(x + y - z = 0\). 64. \(x = y = z = 0\). 65. \(x - y = z + 1 = 0\). 66. \(x - y = y = z - x\). 67. \(x = y = 0\). 68. \(y = z = 1\). 69. \(x + y + z \geq 0\). 70. \(x = y \text{ and } z \geq 0\). 71. \(xy \geq 0\). 72. \(|x| = |y|\). 73. \(xy \geq 0 \text{ and } |x| = |y|\). 74. \(z = x^2 - y^2\). 75. \(z = x^2 + y^2\). 76. \(x^2 + y^2 + z^2 = 1\). 77. \(x(x - 1) = 0\). 78. \(x^2 + 2y^2 + 1 = 0\). 79. \(xy = z = 0\). 80. \(xy = y = 0\).
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