Introductory Linear Algebra/Vectors and subspaces
Vectors[edit | edit source]
Introduction[edit | edit source]
Without distinguishing row and column vectors, we can define vectors as follows:
Definition. (Vector) Let be a positive integer. A vector is an -tuple of real numbers. The set of all such vectors is the Euclidean space of dimension .
- We will define dimension later.
- We use a boldface letter to denote a vector. We may write in written form.
- In particular, we use to denote zero vector in which each entry is zero.
- The entries are called the coordinates or entries of .
A special type of vector is the standard vector.
Definition. (Standard vector) The standard vectors in are whose th entry is and all other entries are , in which
- In , and are usually denoted by and respectively.
- In , , and are usually denoted by , and respectively.
- In , .
- In , .
- In , .
We can see that in is different from in .
However, in linear algebra, we sometimes need to distinguish row and column vectors, which are defined as follows:
Definition. (Row and column vector) A row vector is a matrix, and a column vector is a matrix.
- It is more common to use column vectors.
- Because of this, we can apply the definitions of addition and scalar multiplication of a matrix to the corresponding vector operations.
Example. (Row and column vectors) is a row vector, and is a column vector.
- To save space, we may use to represent column vectors.
- To save more space, it is more common to denote this transpose by .
- On the other hand, we usually do not denote row vectors by to avoid confusion with the notation of vectors (without specifying row or column vectors).
The two basic vector operations are addition and scalar multiplication. Using these two operations only, we can combine multiple vectors as in the following definition.
Definition. (Linear combination) Let . A vector is a linear combination of if
Example. The vector is a linear combination of and , while the vector is not.
On the other hand, since
Another concept that is closely related to linear combinations is span.
Let be a nonempty subset of . The span of , denoted by , is the set of all linear combinations of .
- It follows that contains infinitely many vectors, since there are infinitely many possible linear combinations of such vectors.
- It follows that we can use the vectors belonging in the set to generate each vector in .
- We have a span of a set containing some vector(s), instead of a span of vectors.
Example. The span of the set is
The span of the set is
Linear independence[edit | edit source]
Definition. (Linear (in)dependence)
Let be a subset of . The set is linearly dependent if there exist scalars that are not all zero such that
- (terminology) We may also say that the vectors are linearly independent, instead of saying the set containing these vectors is linearly (in)dependent.
- If the vectors are linearly dependent, it is possible that some of , in the equation, are zero.
- Equivalently, if the vectors are linearly independent, then we have 'if , then the only solution is '.
- This is a more common way to check linear independence.
Then, we will introduce an intuitive result about linear dependence, in the sense that the results match with the name 'linear dependence'.
Proposition. (Equivalent condition for linearly dependence) The vectors are linearly dependent if and only if one of them is a linear combination of the others.
- Only if part:
- without loss of generality, suppose in which (we can replace by another scalar, and the result still holds by symmetry).
- Then, , i.e. is a linear combination of the others.
- If part:
- without loss of generality, suppose (we can similarly replace by another vector, and the result still holds by symmetry).
- Then, .
- Since the coefficient of is nonzero (it is one), are linearly dependent.
- This does not mean that each of them is a linear combination of the others.
- If vectors are 'linearly dependent', we may intuitively think that they are related in a linear sense, and this is true, since one of them is a linear combination of the others, i.e. it has a relationship with all other vectors.
Example. The vectors is linearly dependent.
- Consider the equation .
- Since the determinant of the coefficient matrix
- the coefficient matrix is non-invertible.
- Thus, the homogeneous SLE has a non-trivial solution by simplified invertible matrix theorem, i.e. there exist scalars that are not all zero satisfying the equation.
- the coefficient matrix is non-invertible.
Then, we will introduce a proposition for linearly independent vectors.
Proposition. (Comparing coefficients for linearly independent vectors) Let be linearly independent vectors. If
Example. (Finding unknown coefficients by comparison) Suppose are linearly independent vectors, and
Example. Consider three linearly dependent vectors . Even if
Exercise. Let be linearly independent vectors.
Then, we will discuss two results that relate linear independence with SLE.
Proposition. (Relationship between linear independence and invertibility) Let be a set of vectors in (the number of vectors must be , so that the following matrix is square). Let be the square matrix whose columns are respectively. Then, is linearly independent if and only if is invertible.
Proof. Setting ,
Remark. This gives a convenient way to check linear (in)dependence if the number of vectors involved matches with the number of entries for each of them.
Example. The set is linearly independent, since
Proposition. (Relationship between linear independence and number of solutions in SLE) Let be a SLE ( may not be square matrix, and can be nonzero). If the columns of are linearly independent, then the SLE has at most one solution.
Proof. Let be the columns of , and let .
Then, . Assume there are two distinct solutions and for the SLE, i.e.,
- A special case is that if the columns of are linearly independent, then only has a trivial solution, so is invertible, which matches with the previous proposition.
- The SLE has at most one solution and is equivalent to the SLE having either no solutions or one unique solution.
Example. The set is linearly independent, since none of them is a linear combination of the others. Thus, the SLE
Exercise. Let . It is given that .
Subspaces[edit | edit source]
Then, we will discuss subspaces. Simply speaking, they are some subsets of that have some nice properties. To be more precise, we have the following definition.
Definition. (Subspace) A subset of is a subspace of if all of the following conditions hold.
- (closure under addition) for each ,
- (closure under scalar multiplication) for each and scalar ,
- stands for vector space, since it is a kind of vector space, that is a subset of some larger vector spaces.
- The definition of vector space involves more conditions and is more complicated, and thus not included here.
- For subspaces, after these conditions are satisfied, the remaining conditions for vector spaces are automatically satisfied.
- If is nonempty, then the first condition is redundant.
- But, we may not know whether is empty or nonempty, so it may be more convenient to simply check the first condition.
- This is because for each , by closure under scalar multiplication, and thus by closure under addition.
Example. (Zero space) The set containing only the zero vector, , is a subspace of , and is called the zero space.
- for each scalar .
Example. The span of the set is a subspace of .
Proof. Let .
- since , is a linear combination of and ;
- , since
- let and , then .
- , since
- let , then .
We can see from this example that the entries themselves do not really matter. Indeed, we have the following general result:
Proposition. (a span of finite set is a subspace) For each finite set , is a subspace.
Proof. The idea of the proof is shown in the above example:
- since zero vector is a linear combination of the vectors belonging to ;
- since is a linear combination of the vectors belonging to ;
- since is a linear combination of the vectors belonging to .
In particular, we have special names for some of the spans, as follows:
Definition. (Row, column, and null space) Let be a matrix. The row (column) space of is the span of the rows (columns) of , denoted by (). The null space (or kernel) of is the solution set to the homogeneous SLE , denoted by (or for the name 'kernel').
- It follows from the proposition about the span of a finite set being a subspace, that row and column spaces are subspaces.
- Row and column spaces may belong to different Euclidean spaces,
- e.g. for a matrix , , and .
Example. Null space is a subspace.
Proof. Consider a homogeneous SLE . Let be the solution set to , i.e. .
- since ;
- , because
- since , ;
- since , .
- , because
- since , ;
- since , .
Example. (Example of row, column and null spaces) Consider the matrix .
- (one possible expression)
- since the solution set of is .
Example. The set is a subspace of .
Proof. , so the set is the null space of the matrix, , which is a subspace.
Geometrically, the set is a plane passing through the origin in .
Exercise. Let .
Then, we will introduce some more terminologies related to subspaces.
Definition. (Generating set) Let be a subspace and let be a subset of . The set is a generating set (or spanning set) of if
Definition. (Basis) Let be a subspace. A basis (plural: bases) for is a linearly independent generating set of .
- Basis is quite important, since it tells us the whole structure of , with minimal number of vectors (a generating set of can tell us the whole structure of ).
- The linear independence ensures that there are no 'redundant' vectors in the generating set ('redundant' vectors are the vectors that are linear combinations of the others).
- Given a linearly dependent generating set, we may remove some of the vectors to make it linearly independent (this is known as the reduction theorem, we will discuss this later).
- We usually use to denote basis, since the initial syllable of 'beta' and 'basis' is similar.
The following theorem highlights the importance of basis.
Theorem. Let be a subspace of and let be a subset of . Then, is a basis for if and only if each vector in can be represented as a linear combination of vectors in in a unique way.
- Only if part:
- is a generating set, so each vector in belongs to , i.e. each vector in is a linear combination of vectors in .
- Uniqueness follows from the proposition about comparing coefficients for linear independent vectors.
- If part:
- generates because:
- by definition of subspace, (since linear combinations of vectors in (the vectors in are also in , by ) are belonging to );
- on the other hand, since each vector in can represented as a linear combination of vectors in , we have .
- Thus, we have , so generates by definition.
- is linearly independent because
- let be the vectors in and
- assume in which are not all zero, (i.e. are linearly dependent) then we can express the zero vector in in two different ways:
- which contradicts the uniqueness of representation.
Example. (Standard basis) A basis for is , which is called the standard basis.
Proof. Let .
- generates since
- is linearly independent since
Example. A basis for the set is .
Proof. The general solution to is
There are infinitely many other bases, since we can multiply a vector in this basis by arbitrary nonzero scalar.
Then, we will discuss some ways to construct a basis.
Theorem. (Extension and reduction theorem) Let be a subspace of . Then, the following hold.
- (Extension theorem) Every linearly independent subset of can be extended to a basis for ;
- (Reduction theorem) every finite generating set of consists of a basis for .
Remark. By convention, a basis for the zero space is the empty set .
Its proof is complicated.
Corollary. (Existence of basis for subspace of ) Each subspace of has a basis.
Proof. We start with an empty set . It is linearly independent (since it is not linearly dependent by definition), and is a subset of every set. By extension theorem, it can be extended to a basis for subspace of . Thus, each subspace of has a basis, found by this method.
Example. (Using reduction theorem to find basis) A basis for the subspace is .
Proof. Observe that a generating set of is , since
A terminology that is related to basis is dimension.
Definition. (Dimension) Let be a basis for a subspace of . The number of vectors in , denoted by , is the dimension of .
- By convention, the dimension of the zero space is .
- That is, we say that the number of vectors in empty set is .
- When the subspace has a higher dimension, there is more 'flexibility', since there are more parameters that are changeable.
Recall that there are infinitely many bases for a subspace. Luckily, all bases have the same number of vectors, and so the dimension of subspace is unique, as one will expect intuitively. This is assured bFy the following theorem.
Theorem. (Uniqueness of dimension) Dimension of arbitrary subspace is unique, i.e., if we let and be two finite bases for a subspace of , then, the number of vectors in equals that of .
Proof. Let and . Also, let and be the matrices with 's and 's as columns.
By definition of basis, , and , so
We claim that only has the trivial solution, and this is true, since:
- If , then , and thus , since the columns of are linearly independent, by the proposition about the relationship between linear independence and number of solutions in SLE.
Thus, the RREF of (with columns) has a leading one in each of the first columns. Since there are rows in , we have (if , we cannot have leading ones, since there are at most leading ones)
By symmetry (swapping the role of and ), , and thus
Example. (Dimension of Euclidean space) The dimension of is , since a basis for is standard basis , which contains vectors.
Example. (Dimension of plane) The dimension of the subspace is , since a basis for this subspace is (from a previous example), which contains vectors. Geometrically, the subspace is a plane. In general, the dimension of each plane is .
Then, we will discuss the bases of row, column and null spaces, and also their dimensions.
Proposition. (Basis for row space) Let be a matrix and be the RREF of . Then, a basis for is the set of all nonzero rows of .
Proof. It can be proved that the row space is unchanged when an ERO is performed. E.g.,
- (type I ERO) ;
- (type II ERO) ;
- (type III ERO) .
Assuming this is true, we have . It can be proved that the nonzero rows of generate , and they are linearly independent, so the nonzero rows form a basis for .
Remark. Another basis for is the set of all rows of corresponding to the nonzero rows of , i.e. the set of the rows that are originally located at the position of the nonzero rows of , since it also generates the row space, and is also linear independent.
Example. Let . It can be proved that its RREF is , and therefore a basis for is (it can be proved that is linearly independent), and its dimension is thus .
Another basis is , the corresponding rows to the nonzero rows of the RREF.
Proposition. (Basis for column space) Let be a matrix with columns , and let be RREF of . Suppose columns are the only columns of containing leading ones (they are indexes of column containing leading one). Then, a basis for is .
Proof. Using Gauss-Jordan algorithm, can be transformed to via EROs, and they are row equivalent. Thus, and have the same solution set. Then, it can be proved that linearly (in)dependent columns of correspond to linearly (in)dependent columns of . It follows that is linearly independent, and all other columns belong to the span of this set.
Example. Let . From previous example, the RREF of is . So, the columns of corresponding to the columns of containing leading ones, namely 1st and 2nd columns, form a basis. Thus, a basis for is .
If we let be the 1st, 2nd, 3rd columns of , then . If we let the notations be the corresponding columns of instead, the same equation also holds.
Example. (Basis for null space) Let . From previous example, the RREF of is . A basis for is , since the solution set of is , and its dimension is .
Proposition. (Dimension of null space) Let be a matrix. The dimension of is the number of independent unknowns in the solution set of .
Proof. The idea of the proof is illustrated in the above example: if there are independent unknowns in the solution set, we need a set consisting at least vectors to generate the solution set.
We have special names for the dimensions of row, column and null spaces, as follows:
Definition. (Row rank, column rank and nullity) Let be a matrix. The dimensions of and are called the row rank, column rank, and nullity of . They are denoted by and respectively.
Indeed, the row rank and the column rank of each matrix are the same.
Proposition. (Row and column rank both equal number of leading ones of RREF) For each matrix , is the number of leading ones of the RREF of .
Proof. We can see this from the bases found by the proposition about basis for row space (number of nonzero rows is the number of leading ones of the RREF of ) and the proposition about basis for column space (there are column vectors, and is the number of leading ones by the assumption).
Because of this proposition, we have the following definition.
Definition. (Rank) Let be a matrix. The rank of , denoted by , is the common value of the row rank and the column rank of .
Remark. We usually use this terminology and notation, instead of those for row rank and column rank.
Then, we will introduce an important theorem that relates rank and nullity.
Theorem. (Rank-nullity theorem) Let be an matrix. Then,
Proof. Let be the RREF of . is the number of leading ones of , and is the number of independent unknowns of , which is minus the number of leading ones of . The result follows.
Example. Let . From previous examples,
- A basis for is ;
- a basis for is ;
- a basis for is .
Thus, and . Since the number of columns of is , this verifies the rank-nullity theorem.