2  Vectors and Matrices

Definitions

Vectors

Let \mathbb{F} be a field. The vector vector space \mathbb{F}^{n} is defined as the set of all tuples (ordered lists) that have n field elements of \mathbb{F}

\mathbb{F}^{n} = \left\{ \begin{bmatrix} x_{1} \\ \vdots \\ x_{n} \\ \end{bmatrix}, x_{i} \in \mathbb{F} \right\},

where each element member is called a n dimensional column vector.

  • Vector addition: for \mathbf{x}, \mathbf{y} \in \mathbb{F}^{n}

    \mathbf{x} + \mathbf{y} = \begin{bmatrix} x_{1} + y_{1} \\ \vdots \\ x_{n} + y_{n} \\ \end{bmatrix}.

    That is, the vector addition of the vector space \mathbb{F}^{n} is defined as the element-wise field addition between two vectors.

  • Scalar multiplication: for \alpha \in \mathbb{F} and \mathbf{x} \in \mathbb{F}^{n}

    \alpha \cdot \mathbf{x} = \begin{bmatrix} \alpha \cdot x_{1} \\ \vdots \\ \alpha \cdot x_{n} \\ \end{bmatrix}.

    That is, the scalar multiplication of the vector space \mathbb{F}^{n} is defined as the element-wise field multiplication between the field element and the vector.

  • “zero” vector and addictive inverse: for \mathbf{x} \in \mathbb{F}

    0 = \begin{bmatrix} 0 \\ \vdots \\ 0 \\ \end{bmatrix} \quad \mathbf{x}_{I} = \begin{bmatrix} (x_{1})_{I} \\ \vdots \\ (x_{n})_{I} \\ \end{bmatrix}.

Matrices

Let \mathbb{F} be a field. The matrix vector space \mathbb{F}^{m \times n} is defined as the set of all tables that have m rows and n columns of field elements of \mathbb{F}

\mathbb{F}^{m \times n} = \left\{ \begin{bmatrix} x_{1, 1}, & \dots & x_{1, n} \\ \vdots, & \dots & \vdots \\ x_{m, 1}, & \dots & x_{m, n} \\ \end{bmatrix}, x_{i, j} \in \mathbb{F} \right\},

where each element member is called m \times n dimensional matrix.

The definition of vector addition, scalar multiplication, zero, and addictive inverse are the same as n dimensional column vectors.

Transpose

TODO

Multiplications

Inner product

Given two vectors \mathbf{a} \in \mathbb{F}^{n}, \mathbf{b} \in \mathbb{F}^{n} of the same size, the inner product is a function that maps \mathbf{a} and \mathbf{b} to a field

\mathbf{a} \cdot \mathbf{b} = \mathbf{a}^{T} \mathbf{b} = \sum_{i = 1}^{n} a_{i} b_{i}.

Note that this is a special case of the inner product introduced in Section 6.1 for the vector vector space.

Matrix-vector multiplication

Given a matrix \mathbf{A} \in \mathbb{F}^{m \times n}, the matrix-vector multiplication is a function that maps from vector space \mathbf{x} \in \mathbb{F}^{n} to \mathbf{y} \in \mathbb{F}^{m}

\mathbf{y} = \mathbf{A} \mathbf{x} = \begin{bmatrix} y_{1} = a_{1, 1} \cdot x_{1} + \dots a_{1, n} \cdot x_{n} \\ \vdots \\ y_{m} = a_{m, 1} \cdot x_{1} + \dots a_{m, n} \cdot x_{n} \\ \end{bmatrix}.

If we view \mathbf{A} as n columns of \mathbb{F}^{m} vectors and \mathbf{x} as n coefficients

\mathbf{A} = \begin{bmatrix} | & & | \\ \mathbf{a}_{1} & \dots & \mathbf{a}_{n} \\ | & & | \\ \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x_{1} \\ \vdots \\ x_{n} \\ \end{bmatrix}

the vector \mathbf{y} can be interpreted as the linear combinations of columns of \mathbf{A} with elements of \mathbf{x} as coefficients

\mathbf{y} = \sum_{i=1}^{n} x_{i} \mathbf{a}_{i}.

Matrix multiplication

Given two matrices \mathbf{A} \in \mathbb{F}^{m \times n} and \mathbf{B} \in \mathbb{F}^{n \times r}, the matrix multiplication is a function that returns a matrix by applying vector matrix multiplication on the matrix \mathbf{A} \in \mathbb{F}^{m \times n} and the vector \mathbf{b}_{i} \in \mathbb{F}^{n} (ith column of \mathbf{B})

\mathbf{C} = \mathbf{A} \mathbf{B} = \begin{bmatrix} | & & | \\ \mathbf{c}_{1} & \dots & \mathbf{c}_{r} \\ | & & | \\ \end{bmatrix} = \begin{bmatrix} | & & | \\ \mathbf{A} \mathbf{b}_{1} & \dots & \mathbf{A} \mathbf{b}_{r} \\ | & & | \\ \end{bmatrix}

which results in a vector \mathbf{c}_{i} = \mathbf{A} \mathbf{b}_{i} \in \mathbb{F}^{m} as the ith column of the \mathbf{C}.

  • The column i of \mathbf{C} is a linear combination of columns of \mathbf{A} using the elements of the column i of \mathbf{B} as coefficients.

  • The row i of \mathbf{C} is a linear combination of rows of \mathbf{B} using the elements of row i of \mathbf{A} as coefficients.

Vector-matrix multiplication

Given a vector \mathbf{x} \in \mathbb{F}^{m} and a matrix \mathbf{A} \in \mathbb{F}^{m \times n}, the vector-matrix multiplication is a special case of matrix multiplication between \mathbf{x}^{T} and \mathbf{A}

\mathbf{y}^{T} = \mathbf{x}^{T} \mathbf{A} = \begin{bmatrix} \mathbf{x}^{T} \mathbf{a}_{1} & \dots & \mathbf{x}^{T} \mathbf{a}_{n} \\ \end{bmatrix},

where \mathbf{a}_{i} is the ith column of \mathbf{A}.

The result \mathbf{y} can also be thought as the linear combination of rows of \mathbf{A} using the coefficients in \mathbf{x}

\mathbf{y}^{T} = \sum_{i = 1}^{m} x_{i} \mathbf{A}_{i, *},

where \mathbf{A}_{i, *} is the ith row of the matrix \mathbf{A}.

Outer product

The outer product of two vectors \mathbf{a} \in \mathbb{F}^{m}, \mathbf{b} \in \mathbb{F}^{n} is defined as

\mathbf{a} \otimes \mathbf{b} = \mathbf{a} \mathbf{b}^{T} = \begin{bmatrix} | & & | \\ b_{1} \mathbf{a} & \dots & b_{n} \mathbf{a} \\ | & & | \\ \end{bmatrix}

which is the same as the matrix multiplication by viewing the vector \mathbf{a} as the matrix \mathbf{A} \in \mathbb{F}^{m \times 1} and the row vector \mathbf{b}^{T} as the matrix \mathbb{B} \in \mathbb{F}^{1 \times n}.

Well-known Subspaces for Matrix

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

\mathbf{y} = \mathbf{A} \mathbf{x} = \begin{bmatrix} y_{1} = a_{1, 1} \cdot x_{1} + \dots a_{1, n} \cdot x_{n} \\ \vdots \\ y_{m} = a_{m, 1} \cdot x_{1} + \dots a_{m, n} \cdot x_{n} \\ \end{bmatrix}

is a function that maps from vector space x \in \mathbb{F}^{n} to y \in \mathbb{F}^{m}.

Given a matrix \mathbf{A} \in \mathbb{F}^{m \times n}, we can define several subspaces of \mathbb{F}^{n} or \mathbb{F}^{m} using the matrix-vector multiplication of matrix \mathbf{A}.

Null space

The null space of the matrix \mathbf{A} is the set

N (\mathbf{A}) = \left\{ \mathbf{x} \in \mathbb{F}^{n} \mid \mathbf{A} \mathbf{x} = \mathbf{0} \in \mathbb{F}^{m} \right\},

which is the set of the vectors in \mathbb{F}^{n} that is mapped to 0 \in \mathbb{F}^{m} by matrix \mathbf{A}.

Since \mathbf{A} 0 = 0 \in \mathbb{F}^{n}, \mathbf{x} = 0 \in N (A). Thus N (A) is not empty.

Consider \mathbf{x}_{1}, \mathbf{x}_{2} \in N (A). According to the definition of the N (A), \mathbf{A} \mathbf{x}_{1} = 0 and \mathbf{A} \mathbf{x}_{2} = 0.

Then,

\begin{aligned} \mathbf{A} \mathbf{x}_{1} + \mathbf{A} \mathbf{x}_{2} & = 0 \\ \mathbf{A} (\mathbf{x}_{1} + \mathbf{x}_{2}) & = 0. \end{aligned}

Thus N (\mathbf{A}) is closed under vector addition

\mathbf{x}_{1} + \mathbf{x}_{2} \in N (\mathbf{A}).

Also, for all \alpha \in \mathbb{F},

\begin{aligned} \alpha \cdot \mathbf{A} \mathbf{x}_{1} & = \alpha \cdot 0 \\ \mathbf{A} (\alpha \cdot \mathbf{x}_{1}) & = 0. \end{aligned}

Thus N (\mathbf{A}) is closed under scalar multiplication

\alpha \cdot \mathbf{x}_{1} \in N (\mathbf{A}).

Thus, N (\mathbf{A}) is a non-empty set that is closed for both vector addition and scalar multiplication and thus is a subspace.

Range (image) space

The range (image) space of the matrix \mathbf{A} is the set

R (\mathbf{A}) = \left\{ \mathbf{y} \in \mathbb{F}^{m} \mid \mathbf{y} = \mathbf{A} \mathbf{x}, \forall \mathbf{x} \in \mathbb{F}^{n} \right\},

which is the set of vectors in \mathbb{F}^{m} that can be mapped from \mathbb{F}^{n} by matrix \mathbf{A}.

Since \mathbf{x} = 0 \in \mathbb{F}^{n}, \mathbf{y} = 0 \in R (A). Thus R (A) is not empty.

Consider \mathbf{y}_{1}, \mathbf{y}_{2} \in R (A). According to the definition of the R (A), there exists an \mathbf{x}_{1} \in \mathbb{F}^{m} for \mathbf{y}_{1} and an \mathbf{x}_{2} \in \mathbb{F}^{m} for \mathbf{y}_{2}.

Then,

\begin{aligned} \mathbf{y}_{1} + \mathbf{y}_{2} & = \mathbf{A} \mathbf{x}_{1} + \mathbf{A} \mathbf{x}_{2} \\ & = \mathbf{A} (\mathbf{x}_{1} + \mathbf{x}_{2}) \end{aligned}

Since by the closure under addition property,

\mathbf{x}_{1} + \mathbf{x}_{2} \in \mathbb{F}^{m}

and thus R (\mathbf{A}) is closed under vector addition

\mathbf{y}_{1} + \mathbf{y}_{2} \in R (\mathbf{A}).

Also, for all \alpha \in \mathbb{F},

\begin{aligned} \alpha \cdot \mathbf{y}_{1} & = \alpha \cdot \mathbf{A} \mathbf{x}_{1} \\ & = \mathbf{A} (\alpha \cdot \mathbf{y}_{1}) \end{aligned}

Again, since by the closure under scalar multiplication property,

\alpha \cdot \mathbf{x}_{1} \in \mathbb{F}^{m}, \forall \alpha \in \mathbb{F},

and thus R (\mathbf{A}) is closed under scalar multiplication

\alpha \cdot \mathbf{y}_{1} \in R (\mathbf{A}).

Thus, R (\mathbf{A}) is a non-empty set that is closed for both vector addition and scalar multiplication and thus is a subspace.

Column and row space

The column space of the matrix \mathbf{A} is the set of linear combinations of columns of \mathbf{A}

C (\mathbf{A}) = \left\{ \mathbf{y} \in \mathbb{F}^{m} \mid \mathbf{y} = \sum_{i=1}^{n} \alpha_{i} \cdot \mathbf{a}_{*, i}, \forall \alpha_{i} \in \mathbb{F} \right\},

and the row space of the matrix \mathbf{A} is the set of linear combinations of rows of \mathbf{A}

C (\mathbf{A}^{T}) = \left\{ \mathbf{y} \in \mathbb{F}^{m} \mid \mathbf{y} = \sum_{i=1}^{n} \alpha_{i} \cdot \mathbf{a}_{i, *}^{T}, \forall \alpha_{i} \in \mathbb{F} \right\},

which is the same as the column space of \mathbf{A}^{T}.

By the definition of matrix-vector multiplication, the column space of \mathbf{A} is the same as the range space of \mathbf{A}

C (\mathbf{A}) = \left\{ \mathbf{y} \in \mathbb{F}^{m} \mid \mathbf{y} = \sum_{i=1}^{n} \alpha_{i} \cdot \mathbf{a}_{*, i}, \forall \alpha_{i} \in \mathbb{F} \right\} = \left\{ \mathbf{y} \in \mathbb{F}^{m} \mid \mathbf{y} = \mathbf{A} \mathbf{x}, \forall \mathbf{x} \in \mathbb{F}^{n} \right\} = R (\mathbf{A}).