Span is the set of all possible linear combinations of a given set of vectors in a vector space. It helps define the extent to which a set of vectors can cover or represent other vectors within that space, playing a crucial role in understanding subspaces and dimensionality.
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The span of a single non-zero vector is a line through the origin in the direction of that vector.
The span of two non-parallel vectors in two-dimensional space forms a plane through the origin.
If a set of vectors spans a vector space, it means any vector in that space can be written as a linear combination of those vectors.
The span can be used to determine if vectors are linearly independent; if their span equals the whole space, they are independent.
In finite-dimensional spaces, the dimension is equal to the number of vectors in any basis, which means these vectors must span the space.
Review Questions
How does the concept of span relate to the notion of linear combinations and subspaces?
Span directly connects to linear combinations because it defines all possible outcomes from combining given vectors with scalars. When you consider a set of vectors, their span creates a subspace by including all points that can be reached through these linear combinations. Essentially, the span tells you how far you can go using those vectors alone and what kind of subspace they form together.
In what way does understanding the span of a set of vectors contribute to determining whether those vectors form a basis for a vector space?
To determine if a set of vectors forms a basis, you need to check two conditions: they must be linearly independent, and their span must cover the entire vector space. If you find that the span of these vectors includes all possible vectors in that space, it confirms that these vectors not only provide coverage but also do so without redundancy. Thus, confirming they create a basis.
Evaluate how changing the basis affects the representation of vectors in terms of span and linear combinations.
Changing the basis alters how we represent vectors through their linear combinations because different bases can provide unique perspectives on the same vector space. When you switch bases, you might get different coefficients for representing a vector, but the underlying span remains unchanged. This change is important for coordinate systems and understanding how transformations affect spans and dimensions within those spaces.
A basis is a set of linearly independent vectors that span a vector space, meaning every vector in the space can be represented as a linear combination of the basis vectors.
Dimensionality: Dimensionality refers to the number of vectors in a basis for a vector space, indicating how many independent directions are available within that space.