i. Inner Product $x^T y$

Given vectors $\mathbf{u} = (1, 2)$ and $\mathbf{v} = (-1, -1)$, the standard inner product (dot product) in $\mathbb{R}^2$ is defined as:

$\mathbf{u}^T \mathbf{v} = u_1v_1 + u_2v_2$

Calculating the dot product:

$\mathbf{u}^T \mathbf{v} = (1)(-1) + (2)(-1) = -1 - 2 = -3$

Next, compute the magnitudes of the vectors:

$\|\mathbf{u}\| = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}$ $\|\mathbf{v}\| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}$

The cosine of the angle $\theta$ between the vectors is:

$\cos\theta = \frac{\mathbf{u}^T \mathbf{v}}{\|\mathbf{u}\| \|\mathbf{v}\|} = \frac{-3}{\sqrt{5}\sqrt{2}} = \frac{-3}{\sqrt{10}}$

Thus, the angle $\theta$ is:

$\theta = \cos^{-1}\left(\frac{-3}{\sqrt{10}}\right)$

ii. Inner Product $x^T B y$ where $B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

The matrix $B$ is the identity matrix, so the inner product is equivalent to the standard dot product:

$\mathbf{u}^T B \mathbf{v} = \mathbf{u}^T \mathbf{v} = -3$

Since $B$ is the identity matrix, this inner product yields the same result as part (i).

Checking for Orthogonality and Orthonormality

1. Orthogonality: Vectors $\mathbf{u}$ and $\mathbf{v}$ are orthogonal if their inner product is zero. Since $\mathbf{u}^T \mathbf{v} = -3$, the vectors are not orthogonal.

2. Orthonormality: Vectors are orthonormal if they are orthogonal and each vector has a magnitude of 1. Since the vectors are not orthogonal, they are also not orthonormal.

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Would you like a deeper explanation of any part of the solution, or have any other questions?

Relative Questions:

1. How is the angle between two vectors affected by their magnitudes?
2. What conditions must a set of vectors meet to be considered orthonormal?
3. How would the angle change if the inner product was defined with a different matrix $B$?
4. Can you demonstrate the use of another inner product matrix to find angles between vectors?
5. How does the cosine of the angle relate to the concept of projection of vectors?

Tip:

When checking for orthogonality, always compute the dot product and verify if it's zero to confirm that the vectors are perpendicular.