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Find an orthogonal basis of subspace L, given by system of equations:
\begin{cases} x_1 - x_2 - 3x_3 - 3x_4 = 0 \\ -x_1 - 11x_2 + 3x_3 - 7x_4 + 10x_5 = 0 \end{cases} and basis of subspace $L^+$.

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1 Answer 1

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Note that a subspace $L$ is subspace orthogonal to subspace spanned by $\{(1,-1,-3,-3,0),(-1,-11,3,-7,10)\}$, because you can write down equations in form:

$$\langle (1,-1,-3,-3,0),(x_1,x_2,x_3,x_4,x_5) \rangle=0$$

$$\langle (-1,11,3,-7,10),(x_1,x_2,x_3,x_4,x_5) \rangle=0 $$

Now you should:

  1. Find orthonormal basis of $\text{span}\{(1,-1,-3,-3,0),(-1,-11,3,-7,10)\}$.Let it be $\{v_1,v_2\}$.
  2. Extend this basis to orthonormal basis of $\mathbb{R}^5$:$\{v_1,v_2,v_3,v_4,v_5\}$.
  3. $\{v_3,v_4,v_5\}$ is orthonormal basis of $L$.
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