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Question 1204386: Let a ∈ R^n be a vector. Then {a} is linearly independent if and only if a does not equal 0.
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! es, this statement is true and has several interesting consequences:
1. Foundation of Vector Spaces:
Basis: A set of vectors is called a basis for a vector space if it is both linearly independent and spans the entire space. This statement forms the foundation for understanding bases in vector spaces.
If a single vector 'a' is linearly independent (i.e., not the zero vector), it can form a basis for a one-dimensional subspace within the vector space R^n.
2. Spanning Sets:
A single non-zero vector 'a' spans a one-dimensional subspace (a line) in R^n. This means that any scalar multiple of 'a' can be represented as a linear combination of 'a'.
3. Linear Transformations:
In linear algebra, linear transformations can be represented by matrices. This statement has implications for understanding how linear transformations act on single vectors.
4. Geometric Interpretation:
In geometric terms, a non-zero vector represents a direction in space. Its linear independence signifies that it points in a unique direction, not coinciding with the origin.
In summary:
This seemingly simple statement about the linear independence of a single vector has profound implications for understanding the fundamental concepts of vector spaces, linear transformations, and their geometric interpretations.
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