[See Solution] Consider the vectors v_1=(1,2,3), v_2=(0,1,4), and v_3=(4,0,1) in R^3. Show that they form a basis for R^3. Use the Gram-Schmidt process applied
Question: Consider the vectors \(\mathbf{v}_{1}=(1,2,3), \mathbf{v}_{2}=(0,1,4)\), and \(\mathbf{v}_{3}=(4,0,1)\) in \(\mathbb{R}^{3}\).
- Show that they form a basis for \(\mathbb{R}^{3}\).
- Use the Gram-Schmidt process applied to this basis to find an orthonormal basis \(\left(\mathrm{u}_{1}, \mathrm{u}_{2}, \mathbf{u}_{3}\right)\) for \(\mathbb{R}^{3}\).
- Find the inverse of the matrix \(A\) whose columns are \(\mathrm{u}_{1}, \mathrm{u}_{2}\), and \(\mathrm{u}_{3}\).
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