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Section 2.5 Identifying a Basis (EV5)

Subsection 2.5.1 Class Activities

Activity 2.5.1.

Consider the set of vectors
S={[3โˆ’2โˆ’10],[2411],[0โˆ’16โˆ’5โˆ’3],[1230],[3301]}.
(a)
Express the vector [5201] as a linear combination of the vectors in S, i.e. find scalars such that
[5201]=?[3โˆ’2โˆ’10]+?[2411]+?[0โˆ’16โˆ’5โˆ’3]+?[1230]+?[3301].
(b)
Find a different way to express the vector [5201] as a linear combination of the vectors in S.
(c)
Consider another vector [8675]. Without computing the RREF of another matrix, how many ways can this vector be written as a linear combination of the vectors in S?
  1. Zero.
  2. One.
  3. Infinitely-many.
  4. Computing a new matrix RREF is necessary.

Activity 2.5.2.

Letโ€™s review some of the terminology weโ€™ve been dealing with...
(a)
If every vector in a vector space can be constructed as one or more linear combination of vectors in a set S, we can say...
  1. the set S spans the vector space.
  2. the set S fails to span the vector space.
  3. the set S is linearly independent.
  4. the set S is linearly dependent.
(b)
If the zero vector 0โ†’ can be constructed as a unique linear combination of vectors in a set S (the combination multiplying every vector by the scalar value 0), we can say...
  1. the set S spans the vector space.
  2. the set S fails to span the vector space.
  3. the set S is linearly independent.
  4. the set S is linearly dependent.
(c)
If every vector of a vector space can either be constructed as a unique linear combination of vectors in a set S, or not at all, we can say...
  1. the set S spans the vector space.
  2. the set S fails to span the vector space.
  3. the set S is linearly independent.
  4. the set S is linearly dependent.

Definition 2.5.3.

A basis of a vector space V is a set of vectors S contained in V for which
  1. Every vector in the vector space can be expressed as a linear combination of the vectors in S.
  2. For each vector vโ†’ in the vector space, there is only one way to write it as a linear combination of the vectors in S.
These two properties may be expressed more succintly as the statement "Every vector in V can be expressed uniquely as a linear combination of the vectors in S".

Observation 2.5.4.

In terms of a vector equation, a set S={vโ†’1,โ€ฆ,vโ†’n} is a basis of a vector space if the vector equation
x1v1โ†’+โ‹ฏ+xnvnโ†’=wโ†’
has a unique solution for every vector wโ†’ in the vector space.
Put another way, a basis may be thought of as a minimal set of โ€œbuilding blocksโ€ that can be used to construct any other vector of the vector space.

Activity 2.5.5.

Let S be a basis (Definition 2.5.3) for a vector space. Then...
  1. the set S must both span the vector space and be linearly independent.
  2. the set S must span the vector space but could be linearly dependent.
  3. the set S must be linearly independent but could fail to span the vector space.
  4. the set S could fail to span the vector space and could be linearly dependent.

Definition 2.5.7.

The standard basis of Rn is the set {eโ†’1,โ€ฆ,eโ†’n} where
eโ†’1=[100โ‹ฎ00]eโ†’2=[010โ‹ฎ00]โ‹ฏeโ†’n=[000โ‹ฎ01].
In particular, the standard basis for R3 is {eโ†’1,eโ†’2,eโ†’3}={i^,j^,k^}.

Activity 2.5.8.

Take the RREF of an appropriate matrix to determine if each of the following sets is a basis for R4.
(a)
{[1000],[0100],[0010],[0001]}
  1. A basis, because it both spans R4 and is linearly independent.
  2. Not a basis, because while it spans R4, it is linearly dependent.
  3. Not a basis, because while it is linearly independent, it fails to span R4.
  4. Not a basis, because not only does it fail to span R4, itโ€™s also linearly dependent.
(b)
{[230โˆ’1],[2003],[4302],[โˆ’3013]}
  1. A basis, because it both spans R4 and is linearly independent.
  2. Not a basis, because while it spans R4, it is linearly dependent.
  3. Not a basis, because while it is linearly independent, it fails to span R4.
  4. Not a basis, because not only does it fail to span R4, itโ€™s also linearly dependent.
(c)
{[230โˆ’1],[2003],[313716],[โˆ’110714],[4302]}
  1. A basis, because it both spans R4 and is linearly independent.
  2. Not a basis, because while it spans R4, it is linearly dependent.
  3. Not a basis, because while it is linearly independent, it fails to span R4.
  4. Not a basis, because not only does it fail to span R4, itโ€™s also linearly dependent.
(d)
{[230โˆ’1],[4302],[โˆ’3013],[3615]}
  1. A basis, because it both spans R4 and is linearly independent.
  2. Not a basis, because while it spans R4, it is linearly dependent.
  3. Not a basis, because while it is linearly independent, it fails to span R4.
  4. Not a basis, because not only does it fail to span R4, itโ€™s also linearly dependent.
(e)
{[530โˆ’1],[โˆ’2103],[4513]}
  1. A basis, because it both spans R4 and is linearly independent.
  2. Not a basis, because while it spans R4, it is linearly dependent.
  3. Not a basis, because while it is linearly independent, it fails to span R4.
  4. Not a basis, because not only does it fail to span R4, itโ€™s also linearly dependent.

Activity 2.5.9.

If {vโ†’1,vโ†’2,vโ†’3,vโ†’4} is a basis for R4, that means RREF[vโ†’1vโ†’2vโ†’3vโ†’4] has a pivot in every row (because it spans), and has a pivot in every column (because itโ€™s linearly independent).
What is RREF[vโ†’1vโ†’2vโ†’3vโ†’4]?
RREF[vโ†’1vโ†’2vโ†’3vโ†’4]=[????????????????]

Subsection 2.5.2 Videos

Figure 16. Video: Verifying that a set of vectors is a basis of a vector space

Exercises 2.5.3 Exercises

Subsection 2.5.4 Mathematical Writing Explorations

Exploration 2.5.11.

  • What is a basis for M2,2?
  • What about M3,3?
  • Could we write each of these in a way that looks like the standard basis vectors in Rm for some m? Make a conjecture about the relationship between these spaces of matrices and standard Eulidean space.

Exploration 2.5.12.

Recall our earlier definition of symmetric matrices. Find a basis for each of the following:
  • The space of 2ร—2 symmetric matrices.
  • The space of 3ร—3 symmetric matrices.
  • The space of nร—n symmetric matrices.

Exploration 2.5.13.

Must a basis for the space P2, the space of all quadratic polynomials, contain a polynomial of each degree less than or equal to 2? Generalize your result to polynomials of arbitrary degree.

Subsection 2.5.5 Sample Problem and Solution

Sample problem Example B.1.9.