Section 2.3 Spanning Sets (VS3)
Learning Outcomes
Determine if a set of Euclidean vectors spans
by solving appropriate vector equations.
Subsection 2.3.1 Class Activities
Observation 2.3.1.
Any single non-zero vector/number
Activity 2.3.2.
How many vectors are required to span
Infinitely Many
Activity 2.3.3.
How many vectors are required to span
Infinitely Many
Fact 2.3.4.
At least
Activity 2.3.5.
Choose any vector
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β
Fact 2.3.6.
The set
is inconsistent for some vector
Note that this happens exactly when
Activity 2.3.7.
Consider the set of vectors
(a)
Rewrite this question in terms of the solutions to a vector equation.
(b)
Answer your new question, and use this to answer the original question.
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β
Activity 2.3.8.
Consider the set of third-degree polynomials
and the question βDoes
(a)
Rewrite this question to be about the solutions to a polynomial equation.
(b)
Answer your new question, and use this to answer the original question.
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β
Activity 2.3.9.
Consider the set of matrices
and the question βDoes
(a)
Rewrite this as a question about the solutions to a matrix equation.
(b)
Answer your new question, and use this to answer the original question.
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β
Activity 2.3.10.
Let
is larger than is smaller than
Subsection 2.3.2 Videos
Subsection 2.3.3 Slideshow
Slideshow of activities available athttps://teambasedinquirylearning.github.io/linear-algebra/2022/VS3.slides.html
.Exercises 2.3.4 Exercises
Exercises available athttps://teambasedinquirylearning.github.io/linear-algebra/2022/exercises/#/bank/VS3/
.Subsection 2.3.5 Mathematical Writing Explorations
Exploration 2.3.11.
Construct each of the following, or show that it is impossible:A set of 2 vectors that spans
A set of 3 vectors that spans
A set of 3 vectors that does not span
A set of 4 vectors that spans