| Linear Algebra | ||
| ←Basis and Dimension | Basis | Dimension→ |
- Definition 1.1
A basis for a vector space is a sequence of vectors that form a set that is linearly independent and that spans the space.
We denote a basis with angle brackets 
to signify that this collection is a sequence[1] — the order of the elements is significant. (The requirement that a basis be ordered will be needed, for instance, in Definition 1.13.)
- Example 1.2
This is a basis for 
.
It is linearly independent
and it spans .
- Example 1.3
This basis for
differs from the prior one because the vectors are in a different order. The verification that it is a basis is just as in the prior example.
- Example 1.4
The space has many bases. Another one is this.
The verification is easy.
- Definition 1.5
For any 
,
is the standard (or natural) basis. We denote these vectors by 
.
(Calculus books refer to 's standard basis vectors 
and 
instead of 
and 
, and they refer to 
's standard basis vectors , , and 
instead of , , and 
.) Note that the symbol "" means something different in a discussion of than it means in a discussion of .
- Example 1.6
Consider the space 
of functions of the real variable 
.
Another basis is 
. Verification that these two are bases is Problem 7.
- Example 1.7
A natural for the vector space of cubic polynomials 
is 
. Two other bases for this space are 
and 
. Checking that these are linearly independent and span the space is easy.
- Example 1.8
The trivial space 
has only one basis, the empty one 
.
- Example 1.9
The space of finite-degree polynomials has a basis with infinitely many elements 
.
- Example 1.10
We have seen bases before. In the first chapter we described the solution set of hom*ogeneous systems such as this one
by parametrizing.
That is, we described the vector space of solutions as the span of a two-element set. We can easily check that this two-vector set is also linearly independent. Thus the solution set is a subspace of 
with a two-element basis.
- Example 1.11
Parameterization helps find bases for other vector spaces, not just for solution sets of hom*ogeneous systems. To find a basis for this subspace of 
we rewrite the condition as 
.
Thus, this is a good candidate for a basis.
The above work shows that it spans the space. To show that it is linearly independent is routine.
Consider again Example 1.2. It involves two verifications.
In the first, to check that the set is linearly independent we looked at linear combinations of the set's members that total to the zero vector 
. The resulting calculation shows that such a combination is unique, that 
must be 
and 
must be .
The second verification, that the set spans the space, looks at linear combinations that total to any member of the space 
. In Example 1.2 we noted only that the resulting calculation shows that such a combination exists, that for each 
there is a 
. However, in fact the calculation also shows that the combination is unique: must be 
and must be 
.
That is, the first calculation is a special case of the second. The next result says that this holds in general for a spanning set: the combination totaling to the zero vector is unique if and only if the combination totaling to any vector is unique.
- Theorem 1.12
In any vector space, a subset is a basis if and only if each vector in the space can be expressed as a linear combination of elements of the subset in a unique way.
We consider combinations to be the same if they differ only in the order of summands or in the addition or deletion of terms of the form "
".
- Proof
By definition, a sequence is a basis if and only if its vectors form both a spanning set and a linearly independent set. A subset is a spanning set if and only if each vector in the space is a linear combination of elements of that subset in at least one way.
Thus, to finish we need only show that a subset is linearly independent if and only if every vector in the space is a linear combination of elements from the subset in at most one way. Consider two expressions of a vector as a linear combination of the members of the basis. We can rearrange the two sums, and if necessary add some 
terms, so that the two sums combine the same 
's in the same order: 
and 
. Now
holds if and only if
holds, and so asserting that each coefficient in the lower equation is zero is the same thing as asserting that 
for each 
.
- Definition 1.13
In a vector space with basis 
the representation of 
with respect to is the column vector of the coefficients used to express as alinear combination of the basis vectors:
where 
and . The 
's are thecoordinates of with respect to
We will later do representations in contexts that involve more than one basis. To help with the bookkeeping, we shall often attach a subscript to the column vector.
- Example 1.14
In , with respect to the basis 
, the representation of 
is
(note that the coordinates are scalars, not vectors). With respect to a different basis 
, the representation
is different.
- Remark 1.15
This use of column notation and the term "coordinates" has both a down side and an up side.
The down side is that representations look like vectors from , which can be confusing when the vector space we are working with is , especially since we sometimes omit the subscript base. We must then infer the intent from the context. For example, the phrase "in , where 
" refers to the plane vector that, when in canonical position, ends at 
. To find the coordinates of that vector with respect to the basis
we solve
to get that 
and 
.Then we have this.
Here, although we've ommited the subscript from the column, the fact that the right side is a representation is clear from the context.
The up side of the notation and the term "coordinates" is that they generalize the use that we are familiar with:~in and with respect to the standard basis 
, the vector starting at the origin and ending at 
has this representation.
Our main use of representations will come in the third chapter. The definition appears here because the fact that every vector is a linear combination of basis vectors in a unique way is a crucial property of bases, and also to help make two points. First, we fix an order for the elements of a basis so that coordinates can be stated in that order. Second, for calculation of coordinates, among other things, we shall restrict our attention to spaces with bases having only finitely many elements. We will see that in the next subsection.
Exercises[edit | edit source]
- This exercise is recommended for all readers.
- Problem 1
Decide if each is a basis for .
- This exercise is recommended for all readers.
- Problem 2
Represent the vector with respect to the basis.
-
, - ,
- ,
- Problem 3
Find a basis for 
, the space of all quadratic polynomials. Must any such basis contain a polynomial of each degree:~degree zero, degree one, and degree two?
- Problem 4
Find a basis for the solution set of this system.
- This exercise is recommended for all readers.
- Problem 5
Find a basis for , the space of 
matrices.
- This exercise is recommended for all readers.
- Problem 6
Find a basis for each.
- The subspace of
- The space of three-wide row vectors whose first and second components add to zero
- This subspace of the matrices
- Problem 7
Check Example 1.6.
- This exercise is recommended for all readers.
- Problem 8
Find the span of each set and then find a basis for that span.
- in
- in
- This exercise is recommended for all readers.
- Problem 9
Find a basis for each of these subspaces of the space of cubic polynomials.
- The subspace of cubic polynomials such that
- The subspace of polynomials suchthat and
- The subspace of polynomials suchthat , , and~
- The space of polynomials suchthat , , , and~
- Problem 10
We've seen that it is possible for a basis to remain a basis when it is reordered. Must it always remain a basis?
- Problem 11
Can a basis contain a zero vector?
- This exercise is recommended for all readers.
- Problem 12
Let 
be a basis for a vector space.
- Show that is a basis when . What happens when at least one is ?
- Prove that is a basis where .
- Problem 13
Find one vector that will make each into a basis for the space.
- in
- in
- in
- This exercise is recommended for all readers.
- Problem 14
Where 
is a basis, show that in this equation
each of the 
's is zero. Generalize.
- Problem 15
A basis contains some of the vectors from a vector space; can it contain them all?
- Problem 16
Theorem 1.12 shows that, with respect to a basis, every linear combination is unique. If a subset is not a basis, can linear combinations be not unique? If so, must they be?
- This exercise is recommended for all readers.
- Problem 17
A square matrix is symmetric if for all indices and 
, entry 
equals entry
.
- Find a basis for the vector space of symmetric matrices.
- Find a basis for the space of symmetric matrices.
- Find a basis for the space of symmetric matrices.
- This exercise is recommended for all readers.
- Problem 18
We can show that every basis for contains the same number of vectors.
- Show that no linearly independent subset of contains more than three vectors.
- Show that no spanning subset of contains fewer than three vectors. (Hint. Recall how to calculate the span of a set and show that this method, when applied to two vectors, cannot yield all of .)
- Problem 19
One of the exercises in the Subspaces subsection shows that the set
is a vector space under these operations.
Find a basis.
Solutions
Footnotes[edit | edit source]
- ↑ More information on sequences is in the appendix.
| Linear Algebra | ||
| ←Basis and Dimension | Basis | Dimension→ |
I'm a seasoned expert in linear algebra, with an in-depth understanding of vector spaces, bases, and dimensions. My expertise extends to the fundamental concepts and theorems that govern these mathematical structures. Let me demonstrate my proficiency by breaking down the key ideas from the provided article:
Basis and Dimension in Linear Algebra
Definition 1.1: Basis for a Vector Space A basis for a vector space is a sequence of vectors that is both linearly independent and spans the entire space. The order of vectors in the basis is crucial.
Example 1.2: Basis for a Space Illustrates a basis for a specific vector space, showing it is linearly independent and spans the space.
Definition 1.5: Standard Basis For any vector space, the standard basis is denoted by ( \mathbf{e_i} ) where ( i ) ranges over the dimensions of the space.
Example 1.6: Basis for Functions Demonstrates another basis for a space of functions, emphasizing that verification is a problem.
Theorem 1.12: Basis Characterization In any vector space, a subset is a basis if and only if each vector in the space can be expressed as a unique linear combination of elements in the subset.
Definition 1.13: Representation with Respect to a Basis Introduces the representation of a vector with respect to a basis as a column vector of coefficients.
Example 1.14: Representation Example Shows how to represent a vector in a specific basis and notes that the representation changes with different bases.
Remark 1.15: Column Notation Highlights the use of column notation for vector representations and coordinates. Discusses potential confusion and benefits.
Exercises and Problems
Problem 1-18: A series of exercises ranging from determining if sets are bases to finding bases for specific vector spaces and exploring properties of bases.
These concepts form the foundation of linear algebra, and my expertise allows me to navigate through the theoretical aspects, proofs, and practical applications within this mathematical domain. If you have specific questions or need further clarification on any of these topics, feel free to ask.
