Recompute the Gram matrices for cases (vi)-(viii) in...

Recompute the Gram matrices for cases (vi)-(viii) in Exercise 3.4.22 for the weighted inner product $\langle\mathbf{x}, \mathbf{y}\rangle=x_1 y_1+\frac{1}{2} x_2 y_2+\frac{1}{3} x_3 y_3+\frac{1}{4} x_4 y_4$.

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Key Concepts

Gram Matrix

A Gram matrix is a symmetric matrix whose entries are the inner products of a set of vectors. It encapsulates the geometric relationships between the vectors, such as angles and lengths, and is instrumental in assessing linear independence. The (i,j) entry of the Gram matrix is given by the inner product of the i-th and j-th vectors from the set.

Weighted Inner Product

A weighted inner product modifies the standard inner product by introducing weights to the individual components of the vectors. In this setting, each coordinate product is multiplied by a prescribed weight, which can adjust the relative importance of each component. This approach generalizes the concept of the dot product and allows for flexibility in defining geometries that emphasize certain dimensions over others.

Inner Product Space

An inner product space is a vector space equipped with an inner product, a function that takes two vectors and returns a scalar. This structure provides a notion of length and angle, making it possible to discuss orthogonality and projection within the space. The weighted inner product is an example that fits within this framework, offering a tailored metric for different applications.

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