A subspace test
A nonempty subset of a vector space is a subspace when it is closed under vector addition and scalar multiplication. The nonempty requirement ensures the zero vector is present.
Algebra / English
Subspaces and Span
Professor Dave Explains · YouTube · 5:49
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Understand subspaces and spans through closure tests and concrete linear combinations.
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0:00A subspace test1:30A parametrized line3:00The smallest containing subspaceLearning script
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A nonempty subset of a vector space is a subspace when it is closed under vector addition and scalar multiplication. The nonempty requirement ensures the zero vector is present.
Vectors of the form (x,0,−x) stay in the same form after addition or scalar multiplication. This set is the span of (1,0,−1), a line through the origin.
The span of a collection consists of all its linear combinations. It is a subspace containing the collection, and every other subspace containing those vectors must also contain their span.
Knowledge cards
A parametrized line
Vectors of the form (x,0,−x) stay in the same form after addition or scalar multiplication. This set is the span of (1,0,−1), a line through the origin.
The smallest containing subspace
The span of a collection consists of all its linear combinations. It is a subspace containing the collection, and every other subspace containing those vectors must also contain their span.
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- Span ExplanationAt 0:00
Why this connection?
A nonempty subset of a vector space is a subspace when it is closed under vector addition and scalar multiplication. The nonempty requirement ensures the zero vector is present. Vectors of the form (x,0,−x) stay in the same form after addition or scalar multiplication. This set is the span of (1,0,−1), a line through the origin. The span of a collection consists of all its linear combinations. It is a subspace containing the collection, and every other subspace containing those vectors must also contain their span.
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Meet the concept↗
A nonempty subset of a vector space is a subspace when it is closed under vector addition and scalar multiplication. The nonempty requirement ensures the zero vector is present.
Conditions: The subset must be nonempty.; The subset must be closed under vector addition.; The subset must be closed under scalar multiplication.
YouTubeSubspaces and Span
0:00 – 1:30Watch this moment ↗
Meet the concept↗
The span of a collection consists of all its linear combinations. It is a subspace containing the collection, and every other subspace containing those vectors must also contain their span.
Conditions: The collection consists of one or more vectors.; Linear combinations involve scalar multiplication and vector addition.
YouTubeSubspaces and Span
3:00 – 5:49Watch this moment ↗
Understand why↗
The set is considered a subspace because vectors of the form (x,0,−x) stay in the same form after addition or scalar multiplication. This set is the span of (1,0,−1), a line through the origin.
Conditions: The vectors are of the form (x,0,-x).; The operation is vector addition or scalar multiplication.
YouTubeSubspaces and Span
1:30 – 3:00Watch this moment ↗
Understand why↗
The span is considered the smallest containing subspace because every other subspace containing those vectors must also contain their span. Since the span consists of all linear combinations, any subspace closed under addition and scalar multiplication that includes the original vectors must include all their combinations.
Conditions: The collection of vectors is fixed.; We are comparing subspaces that contain this collection.
YouTubeSubspaces and Span
3:00 – 5:49Watch this moment ↗
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