Mohit & Neha’s Conversation
Professional Development Program: “Viewing Undergraduate Students as Inventors” Task Session #2 Activity
Mohit and Neha go to a linear algebra class together. They usually have group meetings together in college. An excerpt of a conversation can be found below. They propose a statement at the end of the conversation.
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Mohit: For a linearly dependent set of vectors, we can find a vector that can be expressed as a linear combination of other vectors in the set.
Neha: Yeah, this comes directly from the definition of linear dependence.
Mohit: Hmm. But does it not mean that the vector is kind of a superfluous vector?
Neha: Superfluous? You mean it can be ignored from the set?
Mohit: I mean, the vector can be thought of as something like if that is not present in the set, it can still be made available by other vectors.
Neha: Yes, the linear combination of other vectors. Ohh, you mean such superfluous vectors are always present in a linearly dependent set. Interesting! If no vector is superfluous, then it is a linearly independent set.
Mohit: Also, the span of that linearly dependent set and the span of the set without that superfluous vector are the same.
Neha: Thinking about the spanning set of a vector space, there is no vector in the vector space that cannot be made by the vectors in the spanning set.
Mohit: Spanning set of a vector space? Meaning? What is called a spanning set of a vector space?
Neha: I mean a set that can span the entire vector space is called a spanning set of the vector space.
Mohit: Okay, following this terminology and what you were saying means that a spanning set has no shortage. It can make all vectors of the vector space.
Neha: Yeah. I wonder what it would look like for a basis of a vector space. We have both things: no shortage, no superfluous vector.
Mohit: Hmm. It means that if we have a linearly independent set, then there is no superfluous vector, but there can be a shortage.
Neha: Yes, it can happen when that linear independent set is not a basis.
Mohit: And to make it a basis, we need to add more vectors to it so as to eliminate the shortage.
Neha: Hmm. It means that if we are given a linearly independent set, then it can be made a basis by adding some vectors to it. [Prediction]
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Practice Sheet
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