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In Excrcises $1-6,$ the given set is a basis for a subspace $W .$ Use the Gram-Schmidt process to produce an orthogonal basis for $W$ . $$\left[\begin{array}{r}{3} \\ {0} \\ {-1}\end{array}\right],\left[\begin{array}{r}{8} \\ {5} \\ {-6}\end{array}\right]$$

$\left\{ \left[ \begin{array} { c } { 3 } \\ { 0 } \\ { - 1 } \end{array} \right] + \left[ \begin{array} { c } { - 1 } \\ { 5 } \\ { - 3 } \end{array} \right] \right\}$

Calculus 3

Chapter 6

Orthogonality and Least Square

Section 4

The Gram–Schmidt Process

Vectors

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In mathematics, a vector (…

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In Excrcises $1-6,$ the gi…

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Use the Gram-Schmidt proce…

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[M] Use the Gram-Schmidt p…

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in problem one. We have these two victors as a basis for US office space. W We want to use the Grammy. Schmid poses to produce on our stagnant basis for W. The first system is to consider one of these two victors as V one. The only equals 30 minus one. The second step is to get the two. Based on the projection off the second victor, which is here on the first victim, it equals the second victor, minus the projection off the second victim on the first of it way will project on the first victor. Then we have here the one 30 minus one in the nominator. Here the multiply here by a factor. This factor in the nominator we have the dot product between x two and V one. X two is the second victim. It five minus six and we won 30 minus warm. And in the denominator, we have the dot product off the one and the one. Then in the second term, here we have all victories is Vermont except off X two here, which is the second victor. We want to project on on the first victim. Let's calculate it equals 85 minus six minus. We have the dog product is eight multiplied by three plus five, multiplied by zero minus six, multiplied by minus one gives 30. And in the dominator, we have three multiplied by three, which is mine plus zero plus one equals 10 multiplied boy 30 minus one. Then it equals it minus three, which is minus one five minus zero, which is five. And finally we have minus six plus three, which is minus city. And this is V two, then the two victors. First Victor is 30 minus one. And the second victor is minus 15 minus three. This set off these two victors is an orthogonal basis, or signal basis for W, which is the final answer off our property.

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