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Vector spaces and subspaces, linearly dependent and independent sets of vectors, space spanned by a set of vectors, basis of a vector space, sum and intersection space of two vector spaces, coordinate systems in vector spaces, changes in coordinates due to change in basis

Def. Vector space. Any set of n-vectors over a field F which is closed under both addition
and scalar multiplication is called a vector space. If X_{1}, X_{2}, ... ,X_{m} are m n-vectors, the set of
all linear combinations

k_{1}X_{1} + k_{2}X_{2} + ... + k_{m}X_{m}

where k_{1}, k_{2}, ... ,k_{m} are scalars, is a vector space.

Closure under addition. A set of vectors is said to be closed under addition if the sum of any two of them is a vector of the set.

Closure under scalar multiplication. A set of vectors is said to be closed under scalar multiplication if every scalar multiple of a vector of the set is a vector of the set.

Vector spaces and subspaces – examples.

1. Let A and B be any two non-collinear vectors in the x-y plane. Then any other vector X in the
plane can be expressed as a linear combination of vectors A and B. That is there exist numbers
k_{1} and k_{2} such that X = k_{1}A + k_{2}B for any vector X. Conversely, any linear combination of
vectors A and B gives a vector in the x-y plane, Note the closure idea involved. Any vector in
the plane can be obtained as a linear combination of A and B and any linear combination gives
some vector in the plane. It is a closed system. It is a vector space.

2. Let A, B and C be any three non-coplanar vectors in an x-y-z Cartesian coordinate system.
Then any vector in this x-y-z coordinate system can be expressed as a linear combination of A, B
and C. That is, there exist numbers k_{1}, k_{2}, and k_{3} such that X = k_{1}A + k_{2}B + k_{3}C for any vector
X. Conversely, any linear combination of A, B and C gives some vector in the x-y-z coordinate
system. Note the closure idea involved. It is a closed system. It is a vector space. Let us call it
“Vector space Q”.

3. Pass any plane through the origin of an x-y-z Cartesian coordinate system. Denote the plane
by K. Let A and B be any two non-collinear vectors lying in plane K. Then any linear
combination of vectors A and B is a vector lying in plane K (i.e. if c_{1} and c_{2} are any two numbers,
the vector X = c_{1}A + c_{2}B lies in plane K). Moreover, any vector lying in plane K can be
expressed as a linear combination of vectors A and B (i.e. for any vector X in plane K there exist
numbers c_{1} and c_{2} such that X = c_{1}A + c_{2}B ). Note the closure concept involved. It is a closed
system. The totality of all vectors in plane K constitute a vector space. This space represents a
two-dimensional subspace of Vector space Q above.

4. Pass any line through the origin of an x-y-z Cartesian coordinate system. Denote the line by L. Let A be any vector lying in the line. Then any multiple of vector A is a vector lying in line L.. Moreover, any vector lying in line L can be expressed as a multiple of vector A.. Note the closure concept involved. It is a closed system. The totality of all vectors in line L constitute a vector space. This space represents a one-dimensional subspace of Vector space Q above.

Linear combination of vectors. The vector c_{1}x_{1} + c_{2}x_{2} + ... + c_{m}x_{m } with arbitrary
numerical values for the coefficients c_{1}, c_{2}, ... ,c_{m} is called a linear combination of the vectors
x_{1}, x_{2}, ... ,x_{m} .

Linearly dependent and independent sets of vectors. A set of vectors x_{1}, x_{2}, ... ,x_{m}
is said to be linearly dependent if some one of the vectors in the set can be expressed as a linear
combination of one or more of the other vectors in the set. If none of the vectors in the set can be
expressed as a linear combination of any other vectors of the set, then the set is said to be
linearly independent.

Examples from three-dimensional space. To illustrate the concepts let us consider some examples from three dimensional space.

1. Let A, B and C be any three non-coplanar vectors in an x-y-z Cartesian coordinate system. Then any vector in this x-y-z coordinate system can be expressed as a linear combination of A, B and C. However, none of these three vectors A, B and C can be expressed as a linear combination of the other two. The vectors A, B and C constitute a linearly independent set. Now add another vector D to the set. Consider the set A, B, C and D. This set is a dependent set because vector D can be expressed as a linear combination of A, B and C.

2. Pass any plane through the origin of an x-y-z Cartesian coordinate system. Denote the plane by K. Let A and B be any two non-collinear vectors lying in plane K. Then any linear combination of vectors A and B is a vector lying in plane K. However, neither vector A or B can be expressed as a linear combination of the other. The two vectors form a linearly independent set. Now add another vector C, also lying in plane K, to the set. The set A, B and C is a dependent set because vector C can be expressed as a linear combination of A and B.

A necessary and sufficient condition for the linear independence of a set of vectors. There exists an important algebraic criterion, an algebraic test, which can tell us whether a set of vectors is linearly independent or not. That test is given by the following theorem:

Theorem. A necessary and sufficient condition for the set of vectors x_{1}, x_{2}, ... ,x_{m }to be
linearly independent is that

c_{1}x_{1} + c_{2}x_{2} + ... + c_{m}x_{m} = 0

only when all the scalars c_{i} are zero.

What is the reasoning that leads to the assertion of this theorem? Well, a set of vectors x_{1}, x_{2},
... ,x_{m} is linearly dependent if some one of the vectors in the set can be expressed as a linear
combination of one or more of the other vectors in the set, that is if there exists some vector x_{i} in
the set such that

x_{i} = a_{1}x_{j} + a_{2}x_{k} + ...

for one or more vectors x_{j}, x_{k}, etc. of the set. This condition implies that there exists some subset
of vectors x_{i}, x_{j}, x_{k}, etc. within the full set such that

c_{i}x_{i} + c_{j}x_{j} + c_{k}x_{k} + ... = 0

where c_{i} c_{j}, c_{k}, etc. are non-zero. Said differently, a set is linearly dependent if there exist two or
more non-zero c’s for which the following equation holds true:

c_{1}x_{1} + c_{2}x_{2} + ... + c_{m}x_{m} = 0

(i.e. it is possible for the equation to hold true even though not all of the c’s are zero). If there
does not exist two or more non-zero c’s for which it will hold, then the set of vectors is linearly
independent. The case in which only one of the c’s is non-zero is impossible since c_{i}x_{i} = 0 is not
possible if c ≠ 0. Thus the set of vectors is linearly independent if and only if

c_{1}x_{1} + c_{2}x_{2} + ... + c_{m}x_{m} = 0

only when all the scalars c_{i} are zero.

Linear dependence or independence of a set of vectors is determined from the
rank of a matrix formed from them. Consider a matrix formed from m n-vectors with
each vector corresponding to a row in the matrix. If the rank of the matrix is m the set of vectors
is linearly independent. If the rank is less than m the set of vectors is linearly dependent. If the
rank r is less than m then there are exactly r vectors in the set which are linearly independent and
the remaining vectors c_{1}x_{1} + c_{2}x_{2} + ... + c_{m}x_{m} can be expressed as a linear combination of these
r independent vectors.

Space spanned by a set of vectors. Let x_{1}, x_{2}, ... ,x_{m } be a set of n-vectors in n-dimensional space. They may be a linearly independent set or a linearly dependent set of vectors,
it doesn’t matter. The set of all linear combinations of these vectors corresponds either to all of
n-dimensional space or to some subspace of n-dimensional space. The vector space generated by
all linear combinations of x_{1}, x_{2}, ... ,x_{m } is called the subspace spanned by x_{1}, x_{2}, ... ,x_{m }.

Example. Let us illustrate the concept with an example from three-dimensional space.

Pass any plane through the origin of an x-y-z Cartesian coordinate system. Denote the plane by K. Let A, B, C, D and E be five non-collinear vectors lying in plane K. These five vectors form a set that spans plane K, a subspace of three dimensional space. Any linear combination of these vectors lies in plane K and no linear combination lies outside the plane. Now add to this set a vector F which lies outside this plane. The new set of vectors (vectors A, B, C, D, E and F) span all of three-dimensional space, Why? Because the set now contains three linearly independent vectors and three independent vectors will span all of three-dimensional space.

Basis of a vector space. A basis of a vector space is any set of linearly independent vectors that spans the space. Each vector of the space is then a unique linear combination of the vectors of this basis.

Examples.

1. In three-dimensional space any set of three non-coplanar vectors constitute a basis for the space (choose any three non-coplanar vectors and they qualify as a basis). Any vector in the space can be expressed as a linear combination of these basis vectors and, conversely, any linear combination of these three basis vectors lies in three dimensional space. This basis plays a role analogous to a set of coordinate axes (it is viewed as a non-orthogonal set of coordinate axes that serve as a reference frame from which we can express any other vector in the space).

2. Any two non-collinear vectors in three-dimensional space define a plane that constitutes a subspace of three-dimensional space since any linear combination of these two vectors lies in the plane and, conversely, any vector in the plane can be expressed as a linear combination of these two vectors. Thus these two vectors constitute a basis for a two-dimensional subspace of three-dimensional space. Similarly, a single vector in 3-space constitutes a basis for a one-dimensional subspace of 3-space.

3. In two-dimensional space any set of two non-collinear vectors constitute a basis for the space. These two basis vectors than serve as a non-orthogonal reference frame from which any other vector in the space can be expressed.

Dimension of a vector space. The dimension of a vector space is the number of independent vectors required to span the space.

Subspace notation. N-dimensional space V_{n}(F) has embedded in it subspaces of lesser
dimensions. For example, ordinary three-dimensional space has embedded in it two-dimensional
subspaces in the form of planes passing through the origin of the coordinate system and one-dimensional subspaces in the form of lines passing through the origin. A r-dimensional subspace
of V_{n}(F) is denoted by V_{n}^{r}(F). A two-dimensional subspace of ordinary three-dimensional space
V_{3}(R) would, for example, be denoted by V_{3}^{2}(R).

Sum space of two vector spaces. The sum of two vector spaces P and Q is defined as the totality of all vectors x + y where x is in P and y is in Q. This is a vector space and we call it the sum space of P and Q. The sum space can be regarded as the space spanned by the union of the bases of the spaces P and Q.

Intersection space of two spaces. The intersection space of two vector spaces is the set of all vectors that belong to both spaces.

Example. A plane passing through the origin of an x-y-z Cartesian system in ordinary three-dimensional space represents a two-dimensional subspace of three-dimensional space. Consider two planes P and Q passing through the origin which are assumed not to coincide. The sum space of the two planes is the whole three dimensional space and the intersection space is a straight line (the line of their intersection).

Theorem. The dimensions p and q of two given spaces, the dimensions t of their sum and the dimension s of their intersection satisfy the following relation:

p + q = t + s

Coordinate systems in vector spaces. Consider a n-dimensional vector space with basis
vectors A_{1}, A_{2} , ... ,A_{n} so that an arbitrary vector X of the space has a unique representation

X = u_{1}A_{1} + u_{2}A_{2} + ... + u_{n}A_{n}

The vectors A_{1}, A_{2} , ... ,A_{n} are called a coordinate system or reference system in the space and
u_{1}, u_{2}, ... , u_{n} are the coordinates of X with respect to this system. Thus we can call the n-tuple
{u_{1}, u_{2}, ... , u_{n} } the coordinate vector of X relative to the basis {A_{1}, A_{2} , ... , A_{n} } and denote X
by

where A denotes the basis {A_{1}, A_{2} , ... , A_{n} }.

Elementary basis for n-space. The following set of vectors is a special set of vectors
called the elementary basis (E-basis) for n-space [either Real n-space, V_{n}(R) or Complex n-space, V_{n}(C) ]:

E_{1} = [1, 0, 0, ...., 0]^{T}

E_{2} = [0, 1, 0, ...., 0]^{T}

..............................

E_{n} = [0, 0, 0, ...., 1]^{T} .

E-basis Coordinate System. The n-vectors

E_{1} = [1, 0, 0, ...., 0]^{T}

E_{2} = [0, 1, 0, ...., 0]^{T}

..............................

E_{n} = [0, 0, 0, ...., 1]^{T}

are called the elementary n-vectors, unit n-vectors or elementary unit vectors. The
elementary n-vector E_{j}, whose j-th component is 1, is called the j-th elementary n-vector. The
elementary n-vectors E_{1}, E_{2}, ... ,E_{n} constitute an important basis for V_{n}(F). Every vector X = [x_{1},
x_{2}, ... ,x_{n}]^{T} of n-space V_{n}(F), can be expressed uniquely as the sum

X = x_{1}E_{1} + x_{2}E_{2} + ... + x_{n}E_{n}

of the elementary vectors. The components x_{1}, x_{2}, ... ,x_{n} of X are now called the coordinates of
X relative to the E-basis.

Changes in coordinates due to change in basis. Let us consider the problem of how the coordinates of vectors are changed on transition from one basis to another in an

n-dimensional vector space.

Let the original basis be the usual E-basis E_{1}, E_{2}, ... ,E_{n} . Let x_{1}, x_{2}, ... ,x_{n} be the coordinates of
a vector X with respect to the E-basis . Let Z_{1}, Z_{2}, ... ,Z_{n} some arbitrary other basis. Then
there exist unique numbers a_{1}, a_{2}, ... ,a_{n} such that

X = a_{1}Z_{1} + a_{2}Z_{2} + ... + a_{n}Z_{n}

These numbers a_{1}, a_{2}, ... ,a_{n} represent the coordinates of X relative to the Z-basis. Writing

we have X = [ Z_{1}, Z_{2}, ... ,Z_{n} ] X_{Z} = ZX_{Z}

where Z is the matrix [ Z_{1}, Z_{2}, ... ,Z_{n} ] whose columns are the basis vectors Z_{1}, Z_{2}, ... ,Z_{n} .

Thus we have the following result: the coordinates of a vector X with respect to the E-basis are related to its coordinates with respect to some other Z-basis by

X = ZX_{Z}

where matrix Z , whose columns are the new basis vectors Z_{1}, Z_{2}, ... ,Z_{n} , is called the
“matrix of the coordinate transformation”.

Now let us ask what happens to the coordinates of a point if we move from some Z-basis given
by {Z_{1}, Z_{2}, ... ,Z_{n} } to some other W-basis given by {W_{1}, W_{2}, ... ,W_{n} }. We know

X = ZX_{Z}

X = WX_{W}

So

WX_{W} = ZX_{Z}

and

X_{W} = W^{-1}ZX_{Z}

Theorem. If a vector of V_{n}(F) has coordinates X_{Z }and X_{W} relative to bases {Z_{1}, Z_{2}, ... ,Z_{n} }
and {W_{1}, W_{2}, ... ,W_{n} } of V_{n}(F), there exits a nonsingular matrix P, determined solely by the
two bases and given by P = W^{-1}Z, such that X_{W} = PX_{Z} .

References.

Ayres. Matrices (Schaum).

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