Difference between span and basis

In summary: To determine whether or not a set spans a vector space, I was taught to find its determinant and if det|A|=/= 0 then it spans the space.This set spans R3 but isn't linearly independent.{<1, 0, 0>, <0, 1, 0>, <0, 0, 1>, <1, 1, 1>}This set is linearly independent, but doesn't span R3.
  • #1
MarcL
170
2
I'm just having a small trouble understanding the difference ( occurred while I was doing exercise).

A basis is defined as
1)linearly independent
2)spans the space it is found in.

Here is where I get confused:

To determine whether or not a set spans a vector space, I was taught to find its determinant and if det|A|=/= 0 then it spans the space.
I was also taught that if det|a|=/=0 then it isn't coplanar and therefore it is linearly independent ( can also just solve to see if trivial sol'n...)

But then if both det|a=/= it means that it spans and is linearly independent. Therefore, in my head, it comes with the idea that "spans is related to independence"

Anybody got a good way to differentiate both?
 
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  • #2
MarcL said:
To determine whether or not a set spans a vector space, I was taught to find its determinant and if det|A|=/= 0 then it spans the space.

How were you taught to form the matrix whose determinant you take?
 
  • #3
lets say its in r^3, multiply the 3 vectors by a coefficient k1,k2,k3. put the coefficient in a matrix, find the determinant.
 
  • #4
MarcL said:
I'm just having a small trouble understanding the difference ( occurred while I was doing exercise).

A basis is defined as
1)linearly independent
2)spans the space it is found in.
A basis is a set of vectors that is
1)linearly independent
2)spans the space or subspace it is found in.
MarcL said:
Here is where I get confused:

To determine whether or not a set spans a vector space, I was taught to find its determinant and if det|A|=/= 0 then it spans the space.
I was also taught that if det|a|=/=0 then it isn't coplanar and therefore it is linearly independent ( can also just solve to see if trivial sol'n...)
What does "it isn't coplanar" mean? Who is "it"?
MarcL said:
But then if both det|a=/= it means that it spans and is linearly independent. Therefore, in my head, it comes with the idea that "spans is related to independence"
Not necessarily. This set spans R3 but isn't linearly independent.
{<1, 0, 0>, <0, 1, 0>, <0, 0, 1>, <1, 1, 1>}

This set is linearly independent, but doesn't span R3.
{<1, 0, 0>, <0, 1, 0>}

A spanning set for a space or subspace is a set of vectors for which every vector in the space/subspace is a linear combination of the vectors in the spanning set. In my first example above, every vector in R3 can be written as a linear combination of the four vectors in the set.

A set of n vectors {v1, v2, ... , vn} is linearly independent if the only solution to the equation c1 v1 + c2 v2 + ... + cn vn = 0 is the trivial solution (i.e., c1 = c2 = ... = cn = 0).

A basis for a space/subspace is a set of vectors that spans the space/subspace and is a linearly independent set. If the dimension of the space or subspace is n, a spanning set must have at least n vectors in it. A linearly independent set can have at most n vectors in it. A basis is a minimal spanning set.
MarcL said:
Anybody got a good way to differentiate both?
 
Last edited:
  • #5
MarcL said:
lets say its in r^3, multiply the 3 vectors by a coefficient k1,k2,k3. put the coefficient in a matrix, find the determinant.

Can you give a web link to an explanation of this method?

I don't know what you mean by "multiply the 3 vectors by coefficient k1,k2,k3". Do you have the unknowns k1,k2,k3 in the matrix?

If you put the 3 vectors (as n-tuples of given numbers in R^3) in the matrix then you have a square matrix and can take the determinant. But what does the method tell you to do if you have less than 3 vectors ? What does it say to do if you have more than 3 vectors?
 
  • #6
Mark44 said:
A basis is a set of vectors that is
1)linearly independent
2)spans the space or subspace it is found in.
I don't think this is an improvement over the OP's definition. The set ##\{(1,0,0)\}## "is found in" (I can only assume that this means "is a subset of") infinitely many subspaces of ##\mathbb R^3##, but it only spans one of them. If you meant that the set spans the space it spans, then the second statement isn't saying anything.

The OP's definition is fine in my opinion. If I had to change something about it, I would add some clarity by mentioning the space for which the set is supposed to be a basis:

Let ##V## be a vector space. A set ##B\subseteq V## is said to be a basis for ##V## if
(a) ##B## is linearly indendent.
(b) ##B## spans ##V##.

MarcL said:
I'm just having a small trouble understanding the difference
[...]
Anybody got a good way to differentiate both?
Let ##V## be a vector space. Let ##S## be a non-empty subset of ##V##. Let ##W## be a subspace of ##V##. The following statements are equivalent:

(a) ##W## is the intersection of all subspaces of ##V## that has ##S## as a subset.
(b) If ##U## is a subspace of ##V## such that ##S\subseteq U##, then ##W\subseteq U##. (In other words, ##W## is the smallest subspace that contains ##S##).

For each non-empty subset ##S##, we define the span of ##S## as the unique subspace ##W## that satisfies the equivalent conditions above. Different books use different notations for this subspace. Some common ones are ##\operatorname{span} S## and ##\bigvee S##. The set ##S## is said to span ##W##, to generate W, and to be a spanning set for ##W##. (These statements all mean the same thing, that the equivalent conditions above are satisfied).

Let ##V## be a vector space. Let ##S## be a subset of ##V##. The following statements are equivalent.

(a) ##S## is a maximal linearly independent set in ##V##.
(b) ##S## is a minimal spanning set for ##V##.

If you're not familiar with the minimal/maximal terminology, then these statements need to be explained. This is what they mean:

(a) For all ##T\subseteq V##, if ##T## is linearly independent, then ##T\subseteq S##.
(b) For all ##T\subseteq V##, if ##T## spans ##V##, then ##S\subseteq T##.

One book I read used this theorem to define the term "basis". Such a definition would look like this:

Let ##V## be a vector space. A subset ##S\subseteq V## is said to be a basis for ##V## if it satisfies the equivalent conditions of the theorem. (In other words...if it's a minimal spanning set for ##V##, or equivalently, a maximal linearly independent set in ##V##).

MarcL said:
To determine whether or not a set spans a vector space, I was taught to find its determinant and if det|A|=/= 0 then it spans the space.
I was also taught that if det|a|=/=0 then it isn't coplanar and therefore it is linearly independent ( can also just solve to see if trivial sol'n...)

But then if both det|a=/= it means that it spans and is linearly independent. Therefore, in my head, it comes with the idea that "spans is related to independence"
This approach can't be applied to sets like ##\{(1,0,0),(0,1,0)\}\subseteq R^2##, because the matrix ##\begin{pmatrix}1 & 0 & 0\\ 0 & 1 & 0\end{pmatrix}## isn't even square. To get a square matrix when you consider a subset of ##\mathbb R^n##, you have to start with a set with ##n## elements. If you find that the determinant is non-zero, this tells you that you have found a linearly independent set. Since it has ##n## elements, and ##\mathbb R^n## doesn't contain any linearly independent subset with ##n+1## elements, it's a maximal linearly independent set, and therefore a basis.
 
  • #7
MarcL said:
I'm just having a small trouble understanding the difference ( occurred while I was doing exercise).
A basis is defined as
1)linearly independent
2)spans the space it is found in.
Mark44 said:
A basis is a set of vectors that is
1)linearly independent
2)spans the space or subspace it is found in.
Fredrik said:
I don't think this is an improvement over the OP's definition. The set ##\{(1,0,0)\}## "is found in" (I can only assume that this means "is a subset of") infinitely many subspaces of ##\mathbb R^3##, but it only spans one of them. If you meant that the set spans the space it spans, then the second statement isn't saying anything.
All I was doing was to make what the OP (MarcL) wrote somewhat clearer. For instance, "A basis is defined as 1)linearly independent" omits the idea that we're talking about a set of vectors. Much of his wording I left the same. If I had written my own definitions, I would have used a somewhat different wording.
 

Related to Difference between span and basis

What is the definition of "span" and "basis" in mathematics?

Span refers to the set of all possible linear combinations of a given set of vectors. Basis, on the other hand, refers to a set of linearly independent vectors that span a vector space.

What is the difference between a span and a basis?

The main difference between a span and a basis is that a span can contain redundant or dependent vectors, while a basis only contains independent vectors. Additionally, a basis is the smallest set of vectors that can span a vector space, while a span can contain an infinite number of vectors.

How do you determine the span of a set of vectors?

To determine the span of a set of vectors, you can use the linear combination method. This involves taking a linear combination of the vectors and solving for the coefficients that make the equation true. The resulting coefficients will give you the span of the vectors.

Can a set of vectors have multiple bases?

Yes, a set of vectors can have multiple bases. In fact, there can be an infinite number of bases for a given vector space. However, all of these bases will have the same number of vectors, known as the dimension of the vector space.

How are span and basis used in linear algebra?

Span and basis are fundamental concepts in linear algebra and are used to solve systems of linear equations, determine linear independence, and find the dimension of a vector space. They are also important for understanding linear transformations and solving problems in fields such as physics and engineering.

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