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Define a sub-space and provide an example.
A sub-space is a subset of a vector space that is also a vector space itself under the same operations. An example is the set of all vectors in $ ext{R}^3$ that lie in a plane through the origin.
What is the basis of a vector space? Give an example.
A basis of a vector space is a set of linearly independent vectors that span the entire space. For example, the vectors $egin{pmatrix} 1 \ 0 \ 0 \ ext{ and } egin{pmatrix} 0 \ 1 \ 0 \ 0 \ ext{ form a basis for } ext{R}^2.$
Define linearly independent vectors and provide an example.
Linearly independent vectors are vectors that cannot be expressed as a linear combination of one another. An example is the vectors $egin{pmatrix} 1 \ 0 \ 0 \ ext{ and } egin{pmatrix} 0 \ 1 \ 0 \ ext{ in } ext{R}^2.$
What does it mean for vectors to be linearly dependent? Give an example.
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Briefly explain the relationship between sub-space and basis.
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Study Notes
Sub-Space
- A sub-space is a subset of a vector space that is itself a vector space under the same operations as the original vector space.
- It must be closed under addition and scalar multiplication, meaning that the sum of any two vectors in the sub-space is also in the sub-space, and the scalar multiple of any vector in the sub-space is also in the sub-space.
- Example: The set of all 2x2 matrices with zeros on the diagonal is a sub-space of the vector space of all 2x2 matrices.
Basis of a Vector Space
- The basis of a vector space is a set of linearly independent vectors that span the entire vector space.
- This means that any vector in the vector space can be written as a linear combination of the basis vectors.
- The number of vectors in a basis is called the dimension of the vector space.
- Example: The standard basis for the vector space of all 2x2 matrices is { [1 0] [0 0], [0 1] [0 0], [0 0] [1 0], [0 0] [0 1] }
Linearly Independent Vectors
- Linearly independent vectors are a set of vectors that cannot be written as a linear combination of each other.
- In simpler terms, no vector in the set can be expressed as a sum of scalar multiples of the other vectors in the set.
- Example: The vectors [1 0] and [0 1] are linearly independent in R^2.
Linearly Dependent Vectors
- Linearly dependent vectors are a set of vectors where at least one vector can be expressed as a linear combination of the other vectors in the set.
- In simpler terms, one vector can be written as a sum of scalar multiples of the other vectors.
- Example: The vectors [1 0], [0 1], and [1 1] are linearly dependent in R^2 because [1 1] = [1 0] + [0 1].
Sub-Space and Basis Relationship
- A basis for a sub-space is a set of linearly independent vectors that span the sub-space.
- The dimension of the sub-space is equal to the number of vectors in the basis.
- This means that any vector in the sub-space can be written as a linear combination of the basis vectors of the sub-space.
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Description
This quiz covers essential concepts in linear algebra, including definitions of subspaces, bases, and linear independence. You will also explore examples that illustrate these concepts and the relationship between subspaces and bases. Test your understanding of these foundational ideas in vector spaces.
