Intersections and Linear Spans

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In summary, an intersection is a point or set of points where lines or subspaces meet, while a linear span is the set of all possible linear combinations of a given set of vectors. The intersection of two subspaces is a subset of the union of the two subspaces, and the linear span of a set of vectors can be thought of as the intersection of all subspaces that contain those vectors. These concepts are frequently used in fields such as engineering, computer graphics, and data science to solve equations, find common solutions, and create smooth curves and surfaces.
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TrapMuzik
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Okay, so for the problem before this, I proved that L(S ∩ T ) ⊂ L(S ) ∩ L(T ).

For this problem, I have to give an example where L(S ∩ T ) ̸= L(S ) ∩ L(T ).

So I'm thinking that there are going to be elements in L(S) intersect L(T) that are not in the span of S intersect T. In what sort of case would this happen? I'm not sure what direction to go in.

Thanks!
 
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Figured it out! Never mind
 

What is an intersection and how is it defined?

An intersection is a point or set of points where two or more lines, curves, or surfaces meet. In linear algebra, the intersection of two linear subspaces is the set of vectors that belong to both subspaces. It is denoted by the symbol ∩.

What is the difference between intersection and union?

While an intersection is the set of common elements between two or more sets, a union is the combination of all elements from the sets. In terms of linear algebra, the intersection of two subspaces is a subset of the union of the two subspaces.

What is a linear span?

A linear span is the set of all possible linear combinations of a given set of vectors. It is the smallest subspace that contains all the given vectors. In other words, it is the set of all possible solutions to a system of linear equations.

What is the relationship between intersection and linear span?

The linear span of a set of vectors can be thought of as the intersection of all subspaces that contain those vectors. In other words, it is the smallest subspace that contains all the given vectors, just like how an intersection is the smallest set of points that belong to multiple lines or subspaces.

How are intersections and linear spans used in real-world applications?

Intersections and linear spans are used in a variety of fields such as engineering, computer graphics, and data science. They are often used to solve systems of equations, find common solutions to problems, and determine the best fit for a set of data points. For example, in computer graphics, intersections are used to determine where lines or objects overlap, while linear spans are used to create smooth curves and surfaces.

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