Span refers to the set of all possible linear combinations of a given set of vectors. It represents all the points that can be reached in a vector space through these combinations, effectively capturing the extent of coverage these vectors have within that space. The concept of span connects deeply with understanding vector spaces, the relationships between vectors regarding independence and dependence, how coordinates shift during basis changes, and the creation of orthogonal sets in processes like Gram-Schmidt.
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The span of a single non-zero vector is a line through the origin in the direction of that vector.
If you have two non-parallel vectors in three-dimensional space, their span forms a plane through the origin.
The span can be finite or infinite depending on whether the set of vectors is finite or infinite.
A set of vectors that spans a vector space can sometimes be reduced to a smaller set without losing its spanning property, leading to the concept of bases.
If vectors are linearly dependent, at least one vector can be expressed as a linear combination of others, meaning they do not increase the span.
Review Questions
How does the concept of span relate to linear independence and dependence among a set of vectors?
The concept of span is closely linked to linear independence and dependence. A set of vectors is said to be linearly independent if none of the vectors can be expressed as a linear combination of the others. In contrast, if at least one vector can be expressed this way, it indicates linear dependence and suggests that the spanning capability of that set can be reduced without losing coverage in the vector space. Understanding this relationship helps clarify how much 'space' a group of vectors actually covers.
What role does span play when changing coordinate systems and understanding coordinate vectors?
Span plays an essential role when changing coordinate systems because it defines how different bases represent the same vector space. When transitioning from one basis to another, understanding the span helps us determine how coordinate vectors are transformed. Each coordinate system's basis spans the same underlying space, but with different representations. This understanding is crucial for effectively manipulating and translating between different perspectives in linear algebra.
Evaluate how the Gram-Schmidt process utilizes the concept of span to create orthogonal sets from any basis in a vector space.
The Gram-Schmidt process takes a set of linearly independent vectors and systematically transforms them into an orthogonal set while preserving their original span. This means that even though the resulting orthogonal vectors may look different, they still cover the same space as the original set. By maintaining this span during transformation, Gram-Schmidt ensures that we can simplify calculations while still representing any vector in that space as a combination of our new orthogonal basis vectors.
A basis is a set of vectors in a vector space that is both linearly independent and spans the entire space, allowing for unique representation of any vector within that space.
The dimension of a vector space is the number of vectors in a basis for that space, representing the maximum number of linearly independent vectors it can accommodate.