24.18. Gram-Schmidt Orthogonalization

Try these first

1. 1.

   What is the first step in the Gram-Schmidt process?

   Hint

   Think about the length of a vector.

2. 2.

   Define orthonormal vectors.

   Hint

   What properties do the vectors have?

3. 3.

   What does the Gram-Schmidt process achieve?

   • Creates linearly dependent vectors
   • Finds a set of orthonormal vectors
   • Normalizes a single vector

   Hint

   Consider what is produced by the process.

4. 4.

   True or False: The Gram-Schmidt process can only be applied to vectors in $\mathbb{R}^3$.

   • True
   • False

   Hint

   Think about vectors in other spaces too.

5. 5.

   Given the vectors $v_1 = (2, -1, 1)$ and $v_2 = (1, 3, -2)$, apply the Gram-Schmidt process to create the orthonormal basis.

   Hint

   Break down the steps: normalization, projection, and adjust.

6. 6.

   Using the orthonormal basis computed from previous vectors, solve for the coefficients of a given vector $w = (0, 0, 1)$ in that basis.

   Hint

   Remember, the coefficients will tell how much of each unit vector is present.