A weighted Euclidean inner product on R^2 is given for the...

A weighted Euclidean inner product on $R^{2}$ is given for the vectors $\mathbf{u}=\left(u_{1}, u_{2}\right)$ and $\mathbf{v}=\left(v_{1}, v_{2}\right) .$ Find a matrix that generates it.
$$\langle\mathbf{u}, \mathbf{v}\rangle=3 u_{1} v_{1}+5 u_{2} v_{2}$$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Matrix Representation of Inner Products

In finite-dimensional spaces, every inner product can be represented as a bilinear form using a matrix. Specifically, for any vectors u and v, the inner product can be written as u?Av, where A is a matrix that encodes the weighting and interaction between the vector components. This representation simplifies computations and provides a clear link between algebraic operations and geometric interpretations.

Symmetric Positive Definite Matrices

A matrix that generates an inner product must be symmetric and positive definite. This ensures that the inner product is commutative (i.e., the order of the vectors does not change the value) and that it satisfies the condition of being strictly positive for all non-zero vectors, which is essential for maintaining the geometric and analytical properties expected of an inner product.

Weighted Euclidean Inner Product

The weighted Euclidean inner product generalizes the standard dot product by assigning different weights to each coordinate of the vectors. This results in an inner product defined as a sum where each product of corresponding components is multiplied by a predetermined weight, emphasizing or diminishing the contribution of each component in the overall calculation.

Please give Ace some feedback