Weighted Inner Product Examples with Solutions: A Comprehensive Guide

📚 Quick Study Guide

• ⚖️ A weighted inner product generalizes the standard inner product by introducing weights to the components of the vectors.
• 🔢 For vectors $u = (u_1, u_2, ..., u_n)$ and $v = (v_1, v_2, ..., v_n)$ in $\mathbb{R}^n$, and positive weights $w_1, w_2, ..., w_n$, the weighted inner product is defined as: $\langle u, v \rangle_w = w_1u_1v_1 + w_2u_2v_2 + ... + w_nu_nv_n$.
• ➕ It still satisfies the properties of an inner product: linearity, symmetry, and positive definiteness.
• 📏 The weighted norm induced by the weighted inner product is $||u||_w = \sqrt{\langle u, u \rangle_w} = \sqrt{w_1u_1^2 + w_2u_2^2 + ... + w_nu_n^2}$.
• 💡 Choosing different weights can emphasize certain components of the vectors over others. This is very useful in data analysis where some features are more important.

Practice Quiz

1. Which of the following is the correct formula for the weighted inner product of vectors $u = (u_1, u_2)$ and $v = (v_1, v_2)$ with weights $w_1$ and $w_2$?

   1. $ \langle u, v \rangle_w = u_1v_1 + u_2v_2 $
   2. $ \langle u, v \rangle_w = w_1u_1 + w_2v_2 $
   3. $ \langle u, v \rangle_w = w_1u_1v_1 + w_2u_2v_2 $
   4. $ \langle u, v \rangle_w = w_1w_2u_1v_1u_2v_2 $

2. Let $u = (1, 2)$ and $v = (3, 4)$ with weights $w_1 = 2$ and $w_2 = 3$. What is the weighted inner product $\langle u, v \rangle_w$?

   1. 11
   2. 14
   3. 18
   4. 26

3. Which property is NOT a requirement for a valid inner product (weighted or standard)?

   1. Linearity
   2. Symmetry
   3. Positive Definiteness
   4. Non-negativity

4. If all weights are equal to 1, what does the weighted inner product become?

   1. Zero
   2. The standard inner product
   3. The weighted norm
   4. Undefined

5. What is the weighted norm of the vector $u = (2, -1)$ with weights $w_1 = 4$ and $w_2 = 9$?

   1. 5
   2. 7
   3. \(\sqrt{7}\)
   4. \(\sqrt{25}\)

6. In the context of data science, what is a practical use of weighted inner products?

   1. Balancing CPU load
   2. Highlighting important features
   3. Calculating the average temperature
   4. Normalizing image pixels

7. Consider two vectors $u$ and $v$, and weights $w_i > 0$. If $\langle u, v \rangle_w = 0$, what can we say about $u$ and $v$?

   1. $u$ and $v$ are parallel
   2. $u$ and $v$ are orthogonal with respect to the weighted inner product
   3. $u$ and $v$ are identical
   4. $u$ and $v$ are linearly dependent