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In linear algebra, when determining if a set is a subspace of a vector space, it is crucial to check for closure under addition. This property ensures that if you take any two vectors from the set, then adding them results in another vector that is also within the set.

To prove closure under addition, consider two arbitrary vectors from a given set. In our specific example with the set \( H \), vectors are of the form \( \left[ \begin{array}{c} 2t_1 \ 0 \ -t_1 \end{array} \right] \) and \( \left[ \begin{array}{c} 2t_2 \ 0 \ -t_2 \end{array} \right] \). Adding these vectors together, you get:

• Combine each corresponding component: \( 2t_1 + 2t_2 \), \( 0 + 0 \), \( -t_1 - t_2 \).

The result is \( \left[ \begin{array}{c} 2(t_1 + t_2) \ 0 \ -(t_1 + t_2) \end{array} \right] \), which is still in \( H \) because \( t_1 + t_2 \) is a scalar. This verifies that \( H \) is closed under addition, contributing to it being a subspace.

Closure under addition confirms that the combination of any two elements in the set does not produce an outside element. This consistency is what makes vector spaces reliable and predictable.

When evaluating whether a set qualifies as a subspace of a vector space, closure under scalar multiplication must be confirmed. This means that if you multiply any vector from the set by any scalar, the result still lies within the set.

For the set \( H \), vectors are defined by the form \( \left[ \begin{array}{c} 2t \ 0 \ -t \end{array} \right] \). Now, let’s multiply this vector by a scalar \( c \). The result becomes:

• Each component is multiplied by \( c \): \( 2ct \), \( 0 \cdot c = 0 \), \( -ct \).

This gives us the new vector \( \left[ \begin{array}{c} 2ct \ 0 \ -ct \end{array} \right] \), which conforms to the format of \( H \). Thus, \( H \) remains unchanged when applying scalar multiplication, indicating it is closed under this operation.

Closure under scalar multiplication demonstrates that the set is immune to distortions caused by scaling. It confirms that no matter how you stretch or shrink the vectors with scalars, they always retain their subspace identity, enhancing the stability of vector spaces.

In linear algebra, a vector space is a collection of objects, called vectors, that can be added together and multiplied by scalars while satisfying certain rules. These spaces are central to vector algebra and possibility theorems in mathematics.

A vector space must adhere to specific conditions:

• It contains a zero vector, serving as an additive identity.
• Every vector has an additive inverse, ensuring that each can be nullified through addition.

Beyond these, closure under addition and scalar multiplication are crucial properties that must be satisfied.

Considering the example of the set \( H \), these conditions are met. It contains the zero vector because when \( t = 0 \), the vector \( \left[ \begin{array}{c} 0 \ 0 \ 0 \end{array} \right] \) is included in the set. Furthermore, vector addition and scalar multiplication verified through our previous sections confirm \( H \) as a legitimate subspace.

Vector spaces provide a formal framework to handle linear equations and transformations comprehensively. They are fundamental across various scientific and engineering fields, offering tools to understand complex systems with simplicity and coherence.