Math Problem Statement
Solution
The problem is asking us to determine bases for and , where:
- and are subspaces of , spanned by specific vectors.
The subspaces are defined as follows:
To find bases for and , we proceed with the following steps:
Step 1: Find a Basis for
The basis vectors for are: We’ll verify if these two vectors are linearly independent. In this case, they are independent, so they form a basis for .
Step 2: Find a Basis for
The basis vectors for are: We should check if these vectors are linearly independent. It turns out that they are linearly independent, so these three vectors form a basis for .
Step 3: Find a Basis for
To find a basis for , we combine all basis vectors of and : We then determine the linearly independent vectors among them. After applying Gaussian elimination, we find that a basis for consists of:
Step 4: Find a Basis for
To find a basis for , we need to find all vectors that can be expressed as a linear combination of both bases of and . This is equivalent to solving a system that satisfies both sets of linear equations represented by the basis vectors of and . Solving this system reveals that the only common vector is: Thus, a basis for is:
Summary
- Basis for :
- Basis for :
Would you like further details or have any questions?
Related Questions
- How do you determine if a set of vectors is linearly independent?
- What are the steps to find the span of a set of vectors?
- How can we use Gaussian elimination to find bases for vector spaces?
- What are some applications of finding the intersection and sum of subspaces?
- How does the rank-nullity theorem relate to subspace intersections?
Tip
Remember that always consists of vectors that satisfy both subspace conditions, while combines all vectors in either or , filtering out redundancies.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Vector Spaces
Basis of Subspaces
Intersection of Subspaces
Span of Vectors
Formulas
Gaussian elimination for determining linear independence
Definition of vector span
Basis determination through elimination
Theorems
Subspace Sum Theorem
Intersection of Subspaces
Linear Independence
Suitable Grade Level
Undergraduate (University level)