5  Linear Map and Rank

Linear Map

Let \mathcal{U} and \mathcal{V} be vector spaces over the same field \mathbb{F}. A map T: \mathcal{U} \to \mathcal{V} is called a linear map if it satisfies the following two properties for all u, v \in \mathcal{U} and all \alpha \in \mathbb{F}:

  1. Additivity: T(u + v) = T(u) + T(v).

  2. Homogeneity (Scalar Multiplication Compatibility): T(\alpha \cdot v) = \alpha \cdot T(v).

These properties ensure that T preserves the structure of vector addition and scalar multiplication in \mathcal{U} when mapped to \mathcal{V}.

Properties of linear map

Let \mathcal{U} and \mathcal{V} be two vector spaces over the same filed \mathbb{F}, and suppose T is a linear map T: \mathcal{U} \to \mathcal{V}.

  • A linear map must satisfy T (0) = 0.

    We prove by contradiction. Suppose there exists a linear map such that

    T (0) \neq 0.

    Suppose v = 0 and according to the definition of the linear map

    \begin{aligned} T (\alpha \cdot v) & = \alpha \cdot T (v), \forall \alpha \in \mathbb{F} \\ T (0) & = \alpha \cdot T (0), \forall \alpha \in \mathbb{F}. \end{aligned}

    Since T (0) \neq 0, we can divide both sides by T (0)

    \alpha = 1, \forall \alpha \in \mathbb{F},

    which raises a contradiction.

  • There exists a matrix representation of T.

    Suppose \text{dim} (\mathcal{U}) = n and \text{dim} (\mathcal{V}) = m.

    Let \{ u_{1}, \dots, u_{n} \} be a basis of \mathcal{U}, and \{ v_{1}, \dots, v_{m} \} be a basis of \mathcal{V}.

    Suppose any vector x \in \mathcal{U} satisfies that

    x = \sum_{i=1}^{n} \alpha_{i} \cdot u_{i},

    and the mapped vector T (x) \in \mathcal{V} satisfies that

    T (x) = \sum_{j=1}^{m} \beta_{j} \cdot v_{j}.

    Since T (u_{i}) \in \mathcal{V}, we have

    T (u_{i}) = \sum_{j=1}^{m} c_{i, j} \cdot v_{j}, \forall i = 1, \dots, n.

    By the definition of the linear map,

    \begin{aligned} T (x) & = T \left( \sum_{i=1}^{n} \alpha_{i} \cdot u_{i} \right) \\ & = \sum_{i=1}^{n} \alpha_{i} \cdot T (u_{i}). \\ & = \sum_{i=1}^{n} \alpha_{i} \sum_{j=1}^{m} c_{i, j} \cdot v_{j} \\ & = \sum_{j=1}^{m} \left( \sum_{i=1}^{n} \alpha_{i} c_{i, j} \right) \cdot v_{j}. \\ \end{aligned}

    Because of the unique representation property,

    \begin{aligned} T(x) = \sum_{j=1}^{m} \beta_{j} \cdot v_{j} & = \sum_{j=1}^{m} \left( \sum_{i=1}^{n} \alpha_{i} c_{i, j} \right) \cdot v_{j} \\ \beta_{j} & = \sum_{i=1}^{n} \alpha_{i} c_{i, j} \quad \forall j = 1, \dots, m, \end{aligned}

    which can be represented in the matrix form

    \begin{bmatrix} \beta_{1} \\ \vdots \\ \beta_{m} \\ \end{bmatrix} = \begin{bmatrix} c_{1, 1} & \dots & c_{n, 1} \\ \vdots & \dots & \vdots \\ c_{1, m} & \dots & c_{n, m} \\ \end{bmatrix} \begin{bmatrix} \alpha_{1} \\ \vdots \\ \alpha_{n} \\ \end{bmatrix}.

    Hence, given any vector x \in \mathcal{U} that has basis coefficients in a given basis \{ u_{1}, \dots, u_{n} \}

    \mathbf{a} = \begin{bmatrix} \alpha_{1} \\ \vdots \\ \alpha_{n} \\ \end{bmatrix},

    there is a mapped vector T (x) \in \mathcal{V} with basis coefficients in a given basis \{ v_{1}, \dots, v_{n} \}

    \mathbf{b} = \begin{bmatrix} \beta_{1} \\ \vdots \\ \beta_{m} \\ \end{bmatrix},

    where

    \mathbf{b} = \mathbf{C} \mathbf{a}.

Generalization of null space and range space

Since every linear map has a matrix representation, the null space and range space defined using matrix can be redefined using linear map.

Given a map T: \mathcal{U} \to \mathcal{V}, the null space (kernel) is

N (T) = \left\{ x \in \mathcal{U} \mid T (x) = 0 \right\},

and the range space (column space) is

R (T) = \left\{ y \in \mathcal{V} \mid y \in T (x), \forall x \in \mathcal{U} \right\}.

Rank

Consider matrix \mathbf{A} \in \mathbb{F}^{m \times n}, the rank of matrix \mathbf{A} is

\text{rank} (\mathbf{A}) = \text{dim} (R (\mathbf{A})).

Properties of rank

Given a matrix \mathbf{A} \in \mathbb{F}^{m \times n},

  • If \text{rank} (\mathbf{A}) = n, the columns of \mathbf{A} are linearly independent.

    TODO

  • Rank-nullity theorem: \text{rank} (\mathbf{A}) + \text{dim} (N (\mathbf{A})) = n

    Since \mathbf{A} represents a linear transform T: \mathcal{U} \to \mathcal{V} with \mathrm{dim} (\mathcal{U}) = m, \mathrm{dim} (\mathcal{V}) = n the rank-nullity theorem can also be stated as

    \mathrm{dim} (R (T)) + \mathrm{dim} (N (T)) = \mathrm{dim} (\mathcal{U}).

    By assuming that \text{dim} (N (T)) = k, there exists a basis \{ b_{1}, \dots, b_{k} \} of N (T).

    Since by definition of the null space, N (T) is a subspace of \mathcal{U}, then according to the property of the basis, there exists a set of vectors \{ b_{k + 1}, \dots, b_{n} \} in \mathcal{U} such that

    \{ b_{1}, \dots, b_{k}, b_{k + 1}, \dots, b_{n} \}

    is a basis of \mathcal{U}.

    Consider any x \in \mathcal{U},

    \begin{aligned} T (x) & = T \left( \sum_{i=1}^{n} \alpha_{i} b_{i} \right) & [\{ b_{1}, \dots, b_{i} \} \text{ is a basis of } \mathcal{U}] \\ & = \sum_{i=1}^{n} \alpha_{i} T (b_{i}) & [\text{T is linear}] \\ \end{aligned}

    Since \{ b_{1}, \dots, b_{k} \} is a basis of N (T), they are also in N (T). According to the definition of the null space

    T (b_{i}) = 0 \quad i = 1, \dots, k.

    Then we can show that every T (x) is a linear combination of T (b_{k+1}) \dots, T (b_{n}):

    \begin{aligned} T (x) & = \sum_{i=1}^{n} \alpha_{i} T (b_{i}) \\ & = \sum_{i=1}^{k} \alpha_{i} T (b_{i}) + \sum_{i=k+1}^{n} \alpha_{i} T (b_{i}) \\ & = 0 + \sum_{i=k+1}^{n} \alpha_{i} T (b_{i}) \\ & = \sum_{i=k+1}^{n} \alpha_{i} T (b_{i}) \end{aligned}.

    Since the definition of the range space is

    R (T) = \{ y \mid y = T(x), \forall x \in \mathcal{U} \},

    all vectors in R (T) are linear combinations of T (b_{k+1}) \dots, T (b_{n}).

    Also, since b_{k+1}, \dots, b_{n} \in \mathcal{U}, we have

    T (b_{k+1}), \dots, T (b_{n}) \in R (T).

    by the definition of the range space.

    Thus, by the property of span

    R (T) = \text{span} (T (b_{k+1}), \dots, T (b_{n})).

    Then we will show that T (b_{k+1}), \dots, T (b_{n}) are linearly independent by supposing a set of \beta_{k+1}, \dots, \beta_{n} such that

    \begin{aligned} \sum_{i=k+1}^{n} \beta_{i} T (b_{i}) & = 0 \\ T \left( \sum_{i=k+1}^{n} \beta_{i} b_{i} \right) & = 0 & [\text{T is linear}] \\ \sum_{i=k+1}^{n} \beta_{i} b_{i} & \in N (T) & [\text{def of null space}] \\ \end{aligned}

    Since \{ b_{1}, \dots, b_{k} \} is also a basis of the N (T), there exists a set of \gamma_{1}, \dots, \gamma_{k} such that

    \begin{aligned} \sum_{i=k+1}^{n} \beta_{i} b_{i} & = \sum_{i=j}^{k} \gamma_{j} b_{j} \\ \sum_{i=k+1}^{n} \beta_{i} b_{i} + \sum_{j=1}^{k} - \gamma_{j} b_{j} & = 0. \end{aligned}

    Since \{ b_{1}, \dots, b_{n} \} are linearly independent, it must hold that

    \beta_{i} = \gamma_{j} = 0 \quad i = k + 1, \dots, n, j = 1, \dots, k.

    Thus, we have shown that

    \sum_{i=k+1}^{n} \beta_{i} T (b_{i}) = 0 \iff \beta_{i} = 0 \quad i = k + 1, \dots, n,

    which means the vectors T (b_{k + 1}), \dots, T (b_{n}) are linearly independent and therefore is a basis of R (T).

    Finally, since

    • \{ b_{1}, \dots, b_{k} \} is a basis of N (T),

    • \{ b_{k + 1}, \dots, b_{n} \} is a basis of R (T), and

    • \{ b_{1}, \dots, b_{n} \} is a basis of \mathcal{U},

    we have the rank-nullity theorem

    \text{dim} (N (T)) + \text{dim} (R (T)) = \text{dim} (\mathcal{U}).

  • \text{rank} (\mathbf{A}) \leq \min (m, n). If \text{rank} (\mathbf{A}) = \min (m, n), matrix is called a full (row or column) rank matrix.

    According to the rank property

    \text{dim} (N (\mathbf{\mathbf{A}})) = n - \text{rank} (\mathbf{A}).

    Since by definition, the number of vectors in a basis is non-negative

    \text{dim} (N (\mathbf{\mathbf{A}})) \geq 0,

    we have

    \text{rank} (\mathbf{A}) \leq n.

    Also, since R (\mathbf{A}) is a subspace of \mathbb{F}^{m} and the property

    \text{rank} (\mathbf{A}) = \text{dim} (R (\mathbf{A})) \leq m.

    Thus,

    \text{rank} (\mathbf{A}) \leq \min(m, n).

  • \text{rank} (\mathbf{A}) = \text{rank} (\mathbf{A}^{T})

    Suppose \mathbf{A} \in \mathbb{F}^{m \times n}. Let \{ \mathbf{a}_{1}, \dots, \mathbf{a}_{n} \in \mathbb{F}^{m} \} be the columns of \mathbf{A}

    \mathbf{A} = \begin{bmatrix} \mathbf{a}_{1} & \dots & \mathbf{a}_{n} \end{bmatrix}.

    Let \text{rank} (\mathbf{A}) = r.

    By the definition of the range space, the columns of \mathbf{A} are in its range space

    \mathbf{a}_{1}, \dots, \mathbf{a}_{n} \in R (\mathbf{A})

    and thus the columns of \mathbf{A} are linear combinations of any basis of R (\mathbf{A}).

    Suppose there exists a basis of R (\mathbf{A}) as columns of \mathbf{B}

    \mathbf{B} = \begin{bmatrix} \mathbf{b}_{1} & \dots & \mathbf{b}_{r} \end{bmatrix}.

    and a matrix \mathbf{C} \in \mathbb{F}^{r \times n}

    \mathbf{C} = \begin{bmatrix} c_{1, 1} & \dots & c_{n, 1} \\ \vdots & \dots & \vdots \\ c_{1, r} & \dots & c_{n, r} \\ \end{bmatrix}

    such that the columns of \mathbf{A} are linear combinations of the basis \mathbf{B} using the columns of \mathbf{C} as cofficients

    \mathbf{A} = \mathbf{B} \mathbf{C}.

    By the definition of matrix multiplication, the rows of \mathbf{A} are also linear combinations of rows of \mathbf{C} using the rows of \mathbf{B} as coefficients. Thus, by the spanning set property,

    R (\mathbf{A}^{T}) \subseteq R (\mathbf{C}^{T}).

    Then, by the dimension property,

    \begin{aligned} \text{dim} (R (\mathbf{A}^{T})) & \leq \text{dim} (R (\mathbf{C}^{T})) \\ \text{rank} (\mathbf{A}^{T}) & \leq \text{rank} (\mathbf{C}^{T}). \end{aligned}

    Since \mathbf{C}^{T} \in \mathbb{F}^{n \times n}, according to the rank property

    \text{rank} (\mathbf{C}^{T}) \leq r = \text{rank} (\mathbf{A}).

    Thus,

    \text{rank} (\mathbf{A}^{T}) \leq \text{rank} (\mathbf{A}).

    Finally, we can use the same argument to prove

    \text{rank} (\mathbf{A}) \leq \text{rank} (\mathbf{A}^{T}).

    by substituting \mathbf{A} as \mathbf{A}^{T}.

    Since \text{rank} (\mathbf{A}^{T}) \leq \text{rank} (\mathbf{A}) and \text{rank} (\mathbf{A}) \leq \text{rank} (\mathbf{A}^{T}) at the same time, we can prove

    \text{rank} (\mathbf{A}) = \text{rank} (\mathbf{A}^{T}).

  • If \mathbf{A} \in \mathbb{R}^{m \times n}, then

    \text{rank} (\mathbf{A}) = \text{rank} (\mathbf{A}^{T} \mathbf{A}).

    TODO

  • If \mathbf{A} \in \mathbb{R}^{m \times n} and \mathbf{B} \in \mathbb{R}^{m \times m} and \mathbf{C} \in \mathbb{R}^{n \times n} are full rank matrices, then

    \text{rank} (\mathbf{A}) = \text{rank} (\mathbf{B} \mathbf{A}) = \text{rank} (\mathbf{A} \mathbf{C}).

    TODO

Rank and matrix inverse

Suppose \mathbf{A} \in \mathbb{F}^{m \times n}. There exists a left inverse matrix \mathbf{B} \in \mathbb{F}^{n \times m} of \mathbf{A} such that

\mathbf{B} \mathbf{A} = \mathbf{I}_{n \times n},

if and only if \text{rank} (\mathbf{A}) = n.

To prove this, we first prove that there exists a matrix \mathbf{C} such that

\mathbf{A} \mathbf{C} = \mathbf{I}_{m \times m} \iff \mathbf{e}_{1}, \dots \mathbf{e}_{m} \in R (A).

First notice that

\mathbf{I}_{m \times m} = \begin{bmatrix} \mathbf{e}_{1} & \dots & \mathbf{e}_{m} \end{bmatrix}

where \mathbf{e}_{1}, \dots, \mathbf{e}_{m} \in \mathbb{F}^{m}.

Thus, \mathbf{e}_{1}, \dots, \mathbf{e}_{m} is a linear combination of the columns of \mathbf{A}. By the definition of the range space, we can see

\mathbf{e}_{1}, \dots, \mathbf{e}_{m} \in R (\mathbf{A}).

Conversely, since \mathbf{e}_{1}, \dots, \mathbf{e}_{m} \in R (\mathbf{A}), each \mathbf{e}_{i}, i = 1, \dots, m is a linear combination of the columns in \mathbf{A}.

Thus, there exists a set of coefficients c_{j, i}, j = 1, \dots, n for the linear combination of \mathbf{e}_{i}, and the matrix of these coefficients is

\mathbf{C} = \begin{bmatrix} c_{1, 1} & \dots & c_{1, m} \\ \vdots & c_{j, i} & \vdots \\ c_{n, 1} & \dots & c_{n, m} \\ \end{bmatrix}.

Then we prove that

\mathbf{e}_{1}, \dots \mathbf{e}_{m} \in R (\mathbf{A}) \iff \text{rank} (\mathbf{A}) = m.

Again notice that \mathbf{e}_{1}, \dots \mathbf{e}_{m} is the standard basis of \mathbb{F}^{m}.

Since R (\mathbf{A}) \subseteq \mathbb{F}^{m}, all vectors in R (\mathbf{A}) are linear combinations of \mathbf{e}_{1}, \dots \mathbf{e}_{m}.

Since we know that \mathbf{e}_{1}, \dots \mathbf{e}_{m} \in R (\mathbf{A}), according to the spanning set property,

R (\mathbf{A}) = \text{span} (\mathbf{e}_{1}, \dots, \mathbf{e}_{m}).

Since by definition \mathbf{e}_{1}, \dots \mathbf{e}_{m} are linearly independent, the set \mathbf{e}_{1}, \dots, \mathbf{e}_{m} is a basis for R (\mathbf{A}).

According to the cardinality of basis,

\text{rank} (\mathbf{A}) = m.

Conversely, since we know \text{rank} (\mathbf{A}) = m = \text{dim} (\mathbb{F}^{m} and R (\mathbf{A}) \subseteq \mathbb{F}^{m},

R (\mathbf{A}) = \mathbb{F}^{m}.

Since by definition \mathbf{e}_{1}, \dots \mathbf{e}_{m} \in \mathbb{F}^{m},

\mathbf{e}_{1}, \dots \mathbf{e}_{m} \in R (\mathbf{A}).

In conclusion, we have proven that there exists a matrix \mathbf{C} such that

\mathbf{A} \mathbf{C} = \mathbf{I}_{m \times m} \iff \mathbf{e}_{1}, \dots \mathbf{e}_{m} \in R (A) \iff \text{rank} (\mathbf{A}) = m.

There exists a right inverse matrix \mathbf{C} \in \mathbb{F}^{n \times m} of \mathbf{A} such that

\mathbf{A} \mathbf{C} = \mathbf{I}_{m \times m}.

if and only if \text{rank} (\mathbf{A}) = m.

We first notice that

\begin{aligned} (\mathbf{B} \mathbf{A})^{T} & = \mathbf{I}_{n \times n}^{T} \\ \mathbf{A}^{T} \mathbf{B}^{T} & = \mathbf{I}_{n \times n}. \end{aligned}

By applying the proof above, we can prove that

\text{rank} (\mathbf{A}^{T}) = n.

According to the rank property,

\text{rank} (\mathbf{A}^{T}) = \text{rank} (\mathbf{A}) = n.

If \mathbf{A} \in \mathbb{F}^{n \times n} is a square matrix and has full rank (\text{rank}(\mathbf{A}) = n), then \mathbf{A} has a unique inverse matrix \mathbf{A}^{-1} such that

\mathbf{A} \mathbf{A}^{-1} = \mathbf{A}^{-1} \mathbf{A} = \mathbf{I}_{n \times n}.

TODO