To find an orthonormal basis for the column space of a matrix, we can use the Gram-Schmidt process followed by normalization. The column space you're asking about is derived from the matrix formed by the column vectors. Let's clarify the vector or matrix you provided:

You wrote something resembling:
$\mathbf{v} = \begin{bmatrix} -1 \\ -2 \\ -4 \\ 4 \\ -2 \\ 0 \\ 2 \end{bmatrix}$

Let's assume this is a single column vector, and we need to find its orthonormal basis.

Step-by-Step Procedure:

1. Orthonormal Basis: For a single non-zero vector, the orthonormal basis is just the normalized version of that vector.

   First, let's find the norm (magnitude) of the vector $\mathbf{v}$: $\|\mathbf{v}\| = \sqrt{(-1)^2 + (-2)^2 + (-4)^2 + (4)^2 + (-2)^2 + (0)^2 + (2)^2}$ $\|\mathbf{v}\| = \sqrt{1 + 4 + 16 + 16 + 4 + 0 + 4} = \sqrt{45}$ $\|\mathbf{v}\| = 3\sqrt{5}$

2. Normalize the vector: Now, we divide each entry of $\mathbf{v}$ by the norm to get the orthonormal basis.

   $\mathbf{v}_{\text{normalized}} = \frac{1}{3\sqrt{5}} \begin{bmatrix} -1 \\ -2 \\ -4 \\ 4 \\ -2 \\ 0 \\ 2 \end{bmatrix}$ $\mathbf{v}_{\text{normalized}} = \frac{1}{3\sqrt{5}} \begin{bmatrix} -1 \\ -2 \\ -4 \\ 4 \\ -2 \\ 0 \\ 2 \end{bmatrix}$ Simplifying, this gives the orthonormal vector: $\mathbf{v}_{\text{normalized}} = \begin{bmatrix} \frac{-1}{3\sqrt{5}} \\ \frac{-2}{3\sqrt{5}} \\ \frac{-4}{3\sqrt{5}} \\ \frac{4}{3\sqrt{5}} \\ \frac{-2}{3\sqrt{5}} \\ 0 \\ \frac{2}{3\sqrt{5}} \end{bmatrix}$

Conclusion:

The orthonormal basis for the column space of the vector is: $\mathbf{v}_{\text{normalized}} = \frac{1}{3\sqrt{5}} \begin{bmatrix} -1 \\ -2 \\ -4 \\ 4 \\ -2 \\ 0 \\ 2 \end{bmatrix}$

Would you like to explore more details or ask further questions?

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Related Questions:

1. How do you apply the Gram-Schmidt process to multiple vectors?
2. What is the significance of orthonormal bases in vector spaces?
3. Can an orthonormal basis be found for any set of vectors?
4. How does normalization affect the properties of a vector?
5. What are some applications of orthonormal bases in machine learning?

Tip: Always check the linear independence of vectors before finding an orthonormal basis!