91Ó°ÊÓ

Subtract each component of \( \mathbf{u} \) from the corresponding component of \( \mathbf{w} \):\[ \mathbf{w} - \mathbf{u} = (-4, 2, -3, -5, 2) - (5, -1, 0, 3, -3) = (-4-5, 2+1, -3-0, -5-3, 2+3) = (-9, 3, -3, -8, 5). \]

Multiply each component of \( \mathbf{v} \) by 2 and \( \mathbf{u} \) by 3, then add the resulting vectors:\[ 2 \mathbf{v} = 2(-1, -1, 7, 2, 0) = (-2, -2, 14, 4, 0), \]\[ 3 \mathbf{u} = 3(5, -1, 0, 3, -3) = (15, -3, 0, 9, -9), \]\[ 2 \mathbf{v} + 3 \mathbf{u} = (-2+15, -2-3, 14+0, 4+9, 0-9) = (13, -5, 14, 13, -9). \]

First compute \( \mathbf{v} - \mathbf{u} \):\[ \mathbf{v} - \mathbf{u} = (-1, -1, 7, 2, 0) - (5, -1, 0, 3, -3) = (-6, 0, 7, -1, 3). \]Then multiply by 3:\[ 3(\mathbf{v} - \mathbf{u}) = 3(-6, 0, 7, -1, 3) = (-18, 0, 21, -3, 9). \]Now calculate \(-\mathbf{w} + 3(\mathbf{v} - \mathbf{u}) \):\[ -\mathbf{w} = -(-4, 2, -3, -5, 2) = (4, -2, 3, 5, -2), \]\[ -\mathbf{w} + 3(\mathbf{v} - \mathbf{u}) = (4, -2, 3, 5, -2) + (-18, 0, 21, -3, 9) = (-14, -2, 24, 2, 7). \]

Calculate \(-\mathbf{v} + 4\mathbf{u} - \mathbf{w} \):\[ -\mathbf{v} = (1, 1, -7, -2, 0), \]\[ 4\mathbf{u} = 4(5, -1, 0, 3, -3) = (20, -4, 0, 12, -12), \]\[ -\mathbf{v} + 4\mathbf{u} = (1+20, 1-4, -7+0, -2+12, 0-12) = (21, -3, -7, 10, -12), \]Subtract \(\mathbf{w}\):\[ -\mathbf{v} + 4\mathbf{u} - \mathbf{w} = (21, -3, -7, 10, -12) - (-4, 2, -3, -5, 2) = (21+4, -3-2, -7+3, 10+5, -12-2) = (25, -5, -4, 15, -14). \]Finally, multiply by 5:\[ 5(-\mathbf{v} + 4\mathbf{u} - \mathbf{w}) = 5(25, -5, -4, 15, -14) = (125, -25, -20, 75, -70). \]

First find \(3\mathbf{w} + \mathbf{v} \):\[ 3\mathbf{w} = 3(-4, 2, -3, -5, 2) = (-12, 6, -9, -15, 6), \]\[ 3\mathbf{w} + \mathbf{v} = (-12+(-1), 6+(-1), -9+7, -15+2, 6+0) = (-13, 5, -2, -13, 6). \]Now multiply by -2:\[ -2(3\mathbf{w} + \mathbf{v}) = -2(-13, 5, -2, -13, 6) = (26, -10, 4, 26, -12). \]Calculate \(2\mathbf{u} + \mathbf{w} \):\[ 2\mathbf{u} = 2(5, -1, 0, 3, -3) = (10, -2, 0, 6, -6), \]\[ 2\mathbf{u} + \mathbf{w} = (10, -2, 0, 6, -6) + (-4, 2, -3, -5, 2) = (6, 0, -3, 1, -4). \]Finally combine results:\[ -2(3\mathbf{w} + \mathbf{v}) + (2\mathbf{u} + \mathbf{w}) = (26, -10, 4, 26, -12) + (6, 0, -3, 1, -4) = (32, -10, 1, 27, -16). \]

First, find \( \mathbf{w} - 5\mathbf{v} + 2\mathbf{u} \):\[ 5\mathbf{v} = 5(-1, -1, 7, 2, 0) = (-5, -5, 35, 10, 0), \]\[ 2\mathbf{u} = 2(5, -1, 0, 3, -3) = (10, -2, 0, 6, -6), \]\[ \mathbf{w} = (-4, 2, -3, -5, 2), \]Therefore,\[ \mathbf{w} - 5\mathbf{v} + 2\mathbf{u} = (-4, 2, -3, -5, 2) - (-5, -5, 35, 10, 0) + (10, -2, 0, 6, -6) = (-4+5-5, 2+5-2, -3-35+0, -5-10+6, 2+0-6), \]\[ = (11, 5, -38, -9, -4). \]Now, multiply by \( \frac{1}{2} \):\[ \frac{1}{2}(11, 5, -38, -9, -4) = \left(\frac{11}{2}, \frac{5}{2}, -19, -\frac{9}{2}, -2\right). \]Finally, add \( \mathbf{v} \):\[ \left( \frac{11}{2}, \frac{5}{2}, -19, -\frac{9}{2}, -2 \right) + (-1, -1, 7, 2, 0) = \left( \frac{11}{2}-1, \frac{5}{2}-1, -19+7, -\frac{9}{2}+2, -2+0 \right), \]\[ = \left( \frac{9}{2}, \frac{3}{2}, -12, -\frac{5}{2}, -2 \right). \]