Linear algebra basis question

In summary, the conversation discusses the conditions for a set of vectors v_1,...,v_k to be considered a basis of a vector space V. It is stated that if v_1,...,v_k span V and after removing any of the vectors the remaining k-1 vectors do not span V, then v_1,...,v_k are linearly independent and therefore a basis for V. The conversation also touches on the importance of clarifying that v_k is an arbitrary vector and not a named one when proving this statement.
  • #1
jimmycricket
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2

Homework Statement


Let [itex]v_1,...,v_k[/itex] be vectors in a vector space [itex]V[/itex]. If [itex]v_1,...,v_k[/itex] span [itex]V[/itex] and after removing any of the vectors the remaining [itex]k-1[/itex] vectors do not span [itex]V[/itex] then [itex]v_1,...,v_k[/itex] is a basis of [itex]V[/itex]?


Homework Equations





The Attempt at a Solution


If [itex]v_1,...,v_k[/itex] span [itex]V[/itex] but [itex]v_1,...,v_{k-1}[/itex] do not then [itex]v_1,...,v_k[/itex] are linearly independent.
If [itex]v_1,...,v_k[/itex] span [itex]V[/itex] and are linearly independent the [itex]v_1,...,v_k[/itex] is a basis of [itex]V[/itex]
Is this reasoning correct?
 
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  • #2
Yes, your reasoning is correct. If any subset of this set of vectors does not span the vector space, then the original set is independent.
 
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  • #3
If I were writing a proof I would want to emphasize that ##v_{k}## is any arbitrary vector of the set and not a named one.

Personally I would say:
{##{v_{1}, v_{2}, ... v_{k}}##} \ {##{v_{i}}##} is linearly dependent for all i in {1,2,..,k}.

But I'm just being nitpicky.
 
  • #4
jimmycricket said:
If [itex]v_1,...,v_k[/itex] span [itex]V[/itex] but [itex]v_1,...,v_{k-1}[/itex] do not then [itex]v_1,...,v_k[/itex] are linearly independent.

This isn't a mathematical point but given the level of the exercise I would guess you are expected to prove this part (but obviously you are the only one who can know what level of detail is required in your homework)
 

1. What is a basis in linear algebra?

A basis is a set of linearly independent vectors that span a vector space. This means that any vector in the space can be expressed as a unique linear combination of the basis vectors.

2. How do you determine if a set of vectors is a basis?

To determine if a set of vectors is a basis, you can use the following criteria:

  • The vectors must be linearly independent, meaning that no vector in the set can be expressed as a linear combination of the other vectors.
  • The vectors must span the entire vector space, meaning that any vector in the space can be written as a linear combination of the basis vectors.

3. Can a vector space have more than one basis?

Yes, a vector space can have multiple bases. This is because there can be different sets of linearly independent vectors that span the same vector space.

4. How does changing the basis affect linear transformations?

Changing the basis can affect linear transformations by changing the way that vectors are expressed in terms of the new basis vectors. This can result in a different representation of the transformation matrix and can also change the coordinates of a vector in the new basis.

5. What is the difference between a basis and a spanning set?

A basis is a specific type of spanning set that meets the criteria of being linearly independent and spanning the entire vector space. A spanning set, on the other hand, is any set of vectors that span the vector space, but may not necessarily be linearly independent.

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