Math Problem Statement
(1 point) Are the following statements true or false?
False
- If π1 and π2 are subspaces of π π of the same dimension, then π1=π2 .
True 2. If π= span{π’1,π’2,π’3 }, then πππ(π)=3 .
True 3. If the set of vectors π is linearly independent in a subspace π then vectors can be added to π to create a basis for π
False 4. If the set of vectors π spans a subspace π , then vectors can be added to π to create a basis for π
True 5. If the set of vectors π spans a subspace π , then vectors can be removed from π to create a basis for π
Solution
Let's review and evaluate the given statements individually:
-
False:
This statement is indeed false. Two subspaces and of can have the same dimension but still be different. For example, different lines through the origin in can both have dimension 1 but are distinct subspaces. -
True:
This statement is true. If and the vectors are linearly independent, then the dimension of the span (subspace ) is equal to the number of independent vectors, which is 3. -
True:
This is true. If a set of vectors is linearly independent in a subspace , you can always add more vectors (if necessary) to to form a basis for , thereby completing the spanning set. -
False:
This statement is false. If the set of vectors already spans the subspace , then adding more vectors to will not create a basis. In fact, it will introduce redundancy and linear dependence. A basis for requires only the minimal set of linearly independent vectors that span . -
True:
This statement is true. If a set of vectors spans a subspace , you can remove redundant vectors (i.e., linearly dependent vectors) from to create a basis for , which is the smallest set of linearly independent vectors that span the space.
Would you like further clarification on any of these explanations?
Here are 5 related questions to expand your understanding:
- How do you determine if a set of vectors is linearly independent?
- What is the difference between a spanning set and a basis?
- Can two subspaces of different dimensions in be equal?
- Why is linear dependence important when constructing a basis?
- How do you find the dimension of a subspace given a set of vectors?
Tip: When building a basis, focus on identifying the linearly independent vectorsβthis ensures that you span the space without unnecessary redundancy.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Subspaces
Spanning Sets
Basis
Linear Independence
Formulas
Dimension of a subspace: dim(S) = number of linearly independent vectors
Span: S = span{v1, v2, ..., vn}
Theorems
Subspace Theorem
Linear Independence Theorem
Basis and Spanning Set Theorem
Suitable Grade Level
University Level (Introductory Linear Algebra)
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