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The "norm" of a vector is a way of measuring its size or length. It is often referred to as the "magnitude" of the vector.

To find the norm of a vector \( \mathbf{u} = (u_1, u_2, ..., u_n) \), you compute:

\[ \|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2 + ... + u_n^2} \]

This formula involves squaring each component of the vector, summing them, and taking the square root of the sum. The norm is important because it provides a way to quantify how long the vector is in space.

In the example, the vector \( \mathbf{u} = (1, -1, 2, 0) \) has its norm calculated as \( \|\mathbf{u}\| = \sqrt{1^2 + (-1)^2 + 2^2 + 0^2} = \sqrt{6} \). This shows that the length of the vector \( \mathbf{u} \) in four-dimensional space is \( \sqrt{6} \). Similarly, for \( \mathbf{v} = (2, 1, 0, -1) \), the norm \( \|\mathbf{v}\| = \sqrt{6} \) indicates that it also stretches to the same length.

The distance between two vectors is a measure of how far apart they are from each other in space. This distance is determined by computing the norm of their difference.

The formula for the distance \( d(\mathbf{u}, \mathbf{v}) \) between vectors \( \mathbf{u} \) and \( \mathbf{v} \) is:

\[ d(\mathbf{u}, \mathbf{v}) = \|\mathbf{u} - \mathbf{v}\| \]

This means you first find the difference \( \mathbf{u} - \mathbf{v} \), then calculate its norm using the method described earlier.

In the example, the vector difference \( \mathbf{u} - \mathbf{v} \) is \( (1-2, -1-1, 2-0, 0+1) = (-1, -2, 2, 1) \). The distance is then computed as \( \sqrt{(-1)^2 + (-2)^2 + 2^2 + 1^2} = \sqrt{10} \). This value represents the straight-line distance separating vector \( \mathbf{u} \) from vector \( \mathbf{v} \).

The inner product, also called the dot product, is a fundamental operation in linear algebra that combines two vectors to yield a single number. To calculate the inner product of two vectors \( \mathbf{u} = (u_1, u_2, ..., u_n) \) and \( \mathbf{v} = (v_1, v_2, ..., v_n) \), use the formula:

\[ \langle \mathbf{u}, \mathbf{v} \rangle = u_1 v_1 + u_2 v_2 + ... + u_n v_n \]

This operation performs a component-wise multiplication followed by summation of the results. The result represents how numerically similar the two vectors are.

In the provided exercise, the inner product \( \langle \mathbf{u}, \mathbf{v} \rangle \) for \( \mathbf{u} = (1, -1, 2, 0) \) and \( \mathbf{v} = (2, 1, 0, -1) \) is computed as \((1 \times 2) + (-1 \times 1) + (2 \times 0) + (0 \times -1) = 1 \). This indicates that \( \mathbf{u} \) and \( \mathbf{v} \) have a minimal degree of similarity in terms of directionality in the space they occupy.