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a) Find an orthonomal basis of \( U=\operatorname{span}\left\{\left[\begin{array}{l}1 \\ 1 \\ 1 \\ 1\end{array}\right],\left[\begin{array}{c}-2 \\ 4 \\ -2 \\ 4\end{array}\right]\right\} \). b) Recall

Answer 1

To find an orthonormal basis of U, we need to first find a basis of U, and then use the Gram-Schmidt process to orthogonalize and normalize the vectors in the basis.

Step-by-step explanation:

To find an orthonormal basis of U, we need to first find a basis of U, and then use the Gram-Schmidt process to orthogonalize and normalize the vectors in the basis. Let's start by finding the basis of U.

The given vectors are v1 = [1, 1, 1, 1] and v2 = [-2, 4, -2, 4]. Since they are linearly independent, they form a basis for U.

Next, we can use the Gram-Schmidt process to orthogonalize and normalize the basis vectors. Let's denote the orthogonalized basis vectors as u1 and u2.

Step 1: Set u1 = v1.

Step 2: Subtract the projection of v2 onto u1 from v2. Let's denote this projection as p.

Step 3: Set u2 = v2 - p.

Finally, normalize the orthogonalized basis vectors by dividing each vector by its magnitude. This will give us the orthonormal basis vectors.