Subspaces and Span
TLDRThe script discusses the concept of subspaces in vector spaces. A subspace is a smaller set within a vector space that satisfies closure and is itself a vector space. An example is provided using R3 as the vector space and a specific subset S made of vectors with form x, 0, -x. It's shown that S satisfies closure, making it a subspace. Additional key concepts covered include linear combinations, spans of sets of vectors (the smallest subspace containing those vectors), and that the span is always a subspace.
Takeaways
- 😀 A subspace is a smaller set within a vector space that is itself a vector space
- 😊 To check if a set is a subspace, we must verify that it satisfies closure - that scalar multiples and sums of vectors in the set remain in the set
- 🧐 We showed an example subspace of R3 consisting of vectors with components x, 0, -x
- 😎 The span of a set of vectors is the set of all linear combinations of those vectors
- 🤓 The span of any vectors in a vector space forms a subspace of that vector space
- 🤔 The span is the smallest subspace containing that set of vectors
- 👍 Span will play an important role in describing vector spaces
- 😀 Any sum of vectors multiplied by scalars is called a linear combination
- 🧐 The span contains all possible linear combinations of a set of vectors
- 😊 We showed an example of the span of 3 specific vectors from R3
Q & A
What is the definition of a subspace?
-A subspace is a smaller set within a vector space that is itself a vector space.
What do we need to check to determine if a set S is a subspace of vector space V?
-We need to check if S satisfies the properties of closure to determine if it is a subspace of V.
Give an example of a set S that forms a subspace of R3.
-The set S made up of vectors with the form x, 0, -x forms a subspace of R3 because it satisfies the properties of closure.
What is a linear combination?
-A linear combination is any sum of elements of a vector space V multiplied by scalars.
What is the span of a set of vectors?
-The span of a set of vectors is the set of all possible linear combinations of those vectors.
Is the span of any vectors in V always a subspace of V?
-Yes, the span of any vectors in V is always a subspace of V.
Why is the span of a set of vectors the smallest subspace containing those vectors?
-The span is the intersection of all subspaces containing those vectors, so it is the smallest subspace that works.
Give an example of the span of three sample vectors.
-The span of the vectors 2,1,-1; 0,2,2; and -1,-1,-1 has the form 2a-c, a+2b-c, -a+2b-c.
What are the two properties we need to check for closure?
-The two properties of closure we need to check are: 1) multiplying a vector by a scalar gives a vector still in the set, and 2) summing two vectors in the set gives another vector still in the set.
Why is the concept of span significant when describing vector spaces?
-Span plays a key role in completely describing vector spaces because it represents the set of all possible vectors that can be reached from a given set of basis vectors.
Outlines
📚 Defining Subspaces Within Vector Spaces
A subspace is a smaller set within a vector space V that satisfies all the properties of a vector space. To qualify as a subspace, the set must be closed under both scalar multiplication and vector addition. An example subspace S is given consisting of all vectors in R3 of the form x, 0, -x. It's verified that S is closed and therefore a subspace.
📏 Introducing Linear Combinations and Span
A linear combination of vectors v1, v2,...vn is any sum of these vectors multiplied by scalars. The span of a set of vectors is the set of all possible linear combinations. An example shows the span of three specific vectors from R3. Notably, the span of any vectors from a space V is itself a subspace of V.
Mindmap
Keywords
💡vector space
💡subspace
💡closure
💡linear combination
💡span
💡R3
💡scalar
💡vector addition
💡basis
💡dimension
Highlights
A subspace is a smaller set within a vector space that is itself a vector space.
To check if a subset S is a subspace, we only need to verify that it satisfies closure.
We choose a subset S of R3 with vectors having the form x, 0, -x and check if it is closed under vector operations.
We verify S is closed under scalar multiplication and vector addition, so S is a subspace.
The span of a set of vectors is the set of all linear combinations of those vectors.
The span of any vectors in a vector space V is always a subspace of V.
The span is the smallest subspace containing that set of vectors.
Let's look at the span of three specific vectors from R3 as an example.
We write out the linear combination of the vectors with scalar coefficients.
Distributing and adding the vectors shows the span has a specific form.
Understanding subspaces and span is key before further discussing vector spaces.
A subspace needs to satisfy closure under vector operations.
Verifying closure for a subset shows if it is a subspace.
The span represents all possible linear combinations.
Span plays a big role in describing vector spaces.
Transcripts
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