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A vector, denoted by a bold lowercase letter such as $$\mathbf{x}$$, can be defined by explicitly listing its elements. This component-wise representation is an ordered sequence of scalars, commonly written as $$\mathbf{x} = (x_1, x_2, \dots, x_n)$$, where $$n$$ represents the vector's dimensionality.

Component-wise Notation for a Vector

Consider the vector represented as $v = (12, -5, 8, 0)$. What is the dimension of this vector, and what is the value of its second component?

Analyzing Vector Notation

A student represents a 3-dimensional vector `v` with components 4, 9, and 2 using the notation `v = {4, 9, 2}`. Identify the primary error in this representation and explain why it is incorrect for defining a vector.

A vector, such as a $$\frac{d}{2}$$-dimensional vector denoted by $$\mathbf{x}$$, can be represented by explicitly listing its individual scalar components. This component-wise representation is typically written as a sequence, for example, $$\mathbf{x} = (x_1, x_2, \dots, x_{d/2})$$.

Component-wise Notation for a d/2-Dimensional Vector

A 4-dimensional vector, denoted as $v$, has its first component equal to 9, its second component equal to -3, its third component equal to 0, and its fourth component equal to 5. Which of the following correctly represents this vector using component-wise notation?

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