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The concept of "Span" in a vector space relates to the set of all possible vectors that can be formed by taking linear combinations of a given set of vectors. For example, if we have vectors \( \mathbf{u} \) and \( \mathbf{v} \), their span includes every vector that can be constructed as \( c_1 \mathbf{u} + c_2 \mathbf{v} \), where \( c_1 \) and \( c_2 \) are any real numbers. When we say that a vector \( \begin{bmatrix} h \ k \end{bmatrix} \) is in the span of \( \{ \mathbf{u}, \mathbf{v} \} \), it means that there exist some \( c_1 \) and \( c_2 \) such that \( \begin{bmatrix} h \ k \end{bmatrix} \) can be expressed in this way. Understanding span is crucial because it helps determine if a vector can be composed of others, maintaining the properties and dimensions of the vector space.

A "Linear Combination" involves combining vectors together using scalar multiplication followed by vector addition. In simpler terms, it means scaling one or more vectors by constants and then adding the results to produce a new vector. In the case of the vectors \( \mathbf{u} = \begin{bmatrix} 2 \ -1 \end{bmatrix} \) and \( \mathbf{v} = \begin{bmatrix} 2 \ 1 \end{bmatrix} \), a linear combination would be \( c_1 \mathbf{u} + c_2 \mathbf{v} \). This means:

• Multiply \( \mathbf{u} \) by \( c_1 \), resulting in \( c_1 \times \begin{bmatrix} 2 \ -1 \end{bmatrix} = \begin{bmatrix} 2c_1 \ -c_1 \end{bmatrix} \)
• Multiply \( \mathbf{v} \) by \( c_2 \), resulting in \( c_2 \times \begin{bmatrix} 2 \ 1 \end{bmatrix} = \begin{bmatrix} 2c_2 \ c_2 \end{bmatrix} \)
• Add the two results together: \( \begin{bmatrix} 2c_1 \ -c_1 \end{bmatrix} + \begin{bmatrix} 2c_2 \ c_2 \end{bmatrix} = \begin{bmatrix} 2c_1 + 2c_2 \ -c_1 + c_2 \end{bmatrix} \)

Understanding linear combinations is essential for solving vector-based problems and is a foundational concept in linear algebra.

A "System of Equations" refers to a set of equations with multiple variables that you solve simultaneously. It often appears in problems involving linear combinations and spans.In this exercise, forming \( \begin{bmatrix} h \ k \end{bmatrix} \) from a linear combination of \( \mathbf{u} \) and \( \mathbf{v} \) leads to the following system of equations:

• \( 2c_1 + 2c_2 = h \)
• \( -c_1 + c_2 = k \)

These equations represent constraints that the coefficients \( c_1 \) and \( c_2 \) must satisfy to form a specific vector \( \begin{bmatrix} h \ k \end{bmatrix} \). Solving a system of equations typically involves methods like substitution or elimination to find exact values of the variables involved. In our solution, we rearrange the second equation to express one variable in terms of the other and substitute back into the first equation, eventually discovering that real solutions exist for any \( h \) and \( k \). This confirms the span assertion by showing that these values let every vector be a combination of \( \mathbf{u} \) and \( \mathbf{v} \). Understanding how to solve such systems is a key skill in mathematics and engineering applications.