$\text{Coordinate Vectors}$

Any object in a vector space can be expressed algebraically as a linear combination of the vectors in any given basis of the vector space.

In a four dimensional vector space, such a linear combination expression looks like this:

$k_0{u}_0+k_1{u}_1+k_2{u}_2+k_3{u}_3$

where each$k_i$, called a coefficient, is a scalar of some sort and each$u_i$is a basis vector of some basis$u$in a four dimensional vector space. Furthermore, there is a one to one correspondence between the coefficients and the basis vectors, and the linear combination is said to be defined relative to the basis$u$.

$u=\left(\begin{array}{llll}{\text{u}}_0,& {\text{u}}_1,& {\text{u}}_2,& {\text{u}}_3\end{array}\right)$

The one-to-one correspondence between the basis vectors and coefficients are as follows.

$k_0\leftrightarrow u_0$

$k_1\leftrightarrow u_1$

$k_2\leftrightarrow u_2$

$k_3\leftrightarrow u_3$

Each term in the linear combination expression above is the product of a coeffient$k_i$and a corresponding basis vector$u_i$.$$The coeffiecient$k_i$specifies how much the basis vector$u_i$contributes to the object defined by the linear combination. If no contribution is made by a basis vector, the corresponding coefficient is zero.

A linear combination expression is verbose in nature and thus uses more symbols than really necessary to convey the required information. Therefore, a coordinate vector is used to convey the same information, compactly without loss of any information.

A coordinate vector is simply an ordered collection of the coefficients used in the linear combination. The collection is presented as a column, where each element (coefficient) corresponds to a basis vector used in the linear combination, and there is an element for each basis vector.

$\left(\begin{array}{l}{k}_0\\ k_1\\ k_2\\ k_3\end{array}\right)k_i\leftrightarrow u_i$

When a basis vector does not contribute to a linear combination, the corresponding term will be zero and thus may be omitted. If this is the case, the missing term may be imagined to be still present in the linear combination expression as a term with a zero coefficient times the corresponding basis vector. The coordinate vector will contain a zero for such a term.

$\left(\begin{array}{l}{k}_0\\ k_1\\ k_2\\ k_3\end{array}\right)k_i\leftrightarrow u_i$

Because a coordinate vector provides the same information provided by a linear combination, it is said to correspond to that linear combination, and vice versa.

Since a linear combination is said to be defined relative to a basis, the corresponding coordinate vector too is said to be defined relative to that same basis.

To explore this further, consider the vector space of polynomials of degree at most five, denoted by${\mathbb{P}}_5\left(F\right)$.

${\mathbb{P}}_5\left(F\right)$has the ordered basis:

$\left(\begin{array}{llllll}1,& x,& x^2,& x^3,& x^4,& x^5\end{array}\right)$

Any polynomial that resides in this vector space can be expressed as a linear combination$$of the vectors in the ordered basis:

$\left(\begin{array}{llllll}1,& x,& x^2,& x^3,& x^4,& x^5\end{array}\right)\to s_0+s_{1}x+s_2{x}^2+s_3{x}^3+s_4{x}^4+s_5{x}^5$

The corresponding coordinate vector is$\left(\begin{array}{l}{s}_0\\ s_1\\ s_2\\ s_3\\ s_4\\ s_5\end{array}\right)$

Each$s_i$is a real number which can take on any value;$$the last non-zero$s_i$determines the degree of the polynomial.

Because each$s_i$is a real number which can take on any value, the vector space${\mathbb{P}}_5\left(F\right)$contains infinitely many polynomials; they have the following forms (coordinate vectors shown in boxes as row vectors transposed).

$\text{Polynomials of degree}0\text{:}{s}_0$

$\phantom{\text{Polynomials of degree}}\boxed{{\left(\begin{array}{llllll}{s}_0,& 0,& 0,& 0,& 0,& 0\end{array}\right)}^T}$

$\text{Polynomials of degree}1\text{:}{s}_0+s_{1}x\left(s_1\ne 0\right)$

$\phantom{\text{Polynomials of degree}}\boxed{{\left(\begin{array}{llllll}{s}_0,& s_1,& 0,& 0,& 0,& 0\end{array}\right)}^T}$

$\text{Polynomials of degree}2\text{:}{s}_0+s_{1}x+s_2{x}^2\left(s_2\ne 0\right)$

$\phantom{\text{Polynomials of degree}}\boxed{{\left(\begin{array}{llllll}{s}_0,& s_1,& s_2,& 0,& 0,& 0\end{array}\right)}^T}$

$\text{Polynomials of degree}3\text{:}{s}_0+s_{1}x+s_2{x}^2+s_3{x}^3\left(s_3\ne 0\right)$

$\phantom{\text{Polynomials of degree}}\boxed{{\left(\begin{array}{llllll}{s}_0,& s_1,& s_2,& s_3,& 0,& 0\end{array}\right)}^T}$

$\text{Polynomials of degree}4\text{:}{s}_0+s_{1}x+s_2{x}^2+s_3{x}^3+s_4{x}^4\left(s_4\ne 0\right)$

$\phantom{\text{Polynomials of degree}}\boxed{{\left(\begin{array}{llllll}{s}_0,& s_1,& s_2,& s_3,& s_4,& 0\end{array}\right)}^T}$

$\text{Polynomials of degree}5\text{:}{s}_0+s_{1}x+s_2{x}^2+s_3{x}^3+s_4{x}^4+s_5{x}^5\left(s_5\ne 0\right)$

$\phantom{\text{Polynomials of degree}}\boxed{{\left(\begin{array}{llllll}{s}_0,& s_1,& s_2,& s_3,& s_4,& s_5\end{array}\right)}^T}$

Note that, to save space, coordinate vectors above are written as not column vectors but row vectors transponsed.

Two concrete examples follow.

$\text{Example 1}$Polynomials of degree three and their coordinate vectors in${\mathbb{P}}_5\left(F\right)$look like this:

$$$s_0+s_{1}x+s_2{x}^2+s_3{x}^3\left(s_3\ne 0\right)$

${\left(\begin{array}{llllll}{s}_0,& s_1,& s_2,& s_3,& 0,& 0\end{array}\right)}^T=\left(\begin{array}{l}{s}_0\\ s_1\\ s_2\\ s_3\\ 0\\ 0\end{array}\right)\left(s_3\ne 0\right)$

The one-to-one correspondence between the basis vectors and components of coordinate vectors:

$s_0\leftrightarrow 1$

$s_1\leftrightarrow x$

$s_2\leftrightarrow x^2$

$s_3\leftrightarrow x^3$

Each coefficient$s_i$(a real number) used in a linear combination expression appears as a component in the corresponding coordinate vector, where components correspond to unique basis vectors.

Here is a random polynomial of degree three in${\mathbb{P}}_5\left(F\right)$and its coordinate vector.

$1+x-x^2-2x^3\leftrightarrow \left(\begin{array}{c}1\\ 1\\ -1\\ -2\\ 0\\ 0\end{array}\right)$

$\text{Example 2}$Polynomials of degree five and their coordinate vectors in${\mathbb{P}}_5\left(F\right)$look like this:

$$$s_0+s_{1}x+s_2{x}^2+s_3{x}^3+s_4{x}^4+s_5{x}^5\left(s_5\ne 0\right)$

${\left(\begin{array}{llllll}{s}_0,& s_1,& s_2,& s_3,& s_4,& s_5\end{array}\right)}^T=\left(\begin{array}{l}{s}_0\\ s_1\\ s_2\\ s_3\\ s_4\\ s_5\end{array}\right)\left(s_5\ne 0\right)$

The one-to-one correspondence between the basis vectors and components of coordinate vectors:

$s_0\leftrightarrow 1$

$s_1\leftrightarrow x$

$s_2\leftrightarrow x^2$

$s_3\leftrightarrow x^3$

$s_4\leftrightarrow x^4$

$s_5\leftrightarrow x^5$

Here is a random polynomial of degree five in${\mathbb{P}}_5\left(F\right)$and its coordinate vector.

$1+x+2x^2-3x^3+2x^4+2x^5\leftrightarrow \left(\begin{array}{c}1\\ 1\\ 2\\ -3\\ 2\\ 2\end{array}\right)$

Each coefficient$s_i$(a real number) specifies how much contribution the corresponding basis vector makes to the linear combination.

A zero coefficient means no contribution is made by that basis vector, and the corresponding term in the linear combination expression is normally not shown. However, there is no harm in imagining that the term is still there with a zero coefficient.

$3+5x^3+7x^5\leftrightarrow \left(\begin{array}{c}3\\ 0\\ 0\\ 5\\ 0\\ 7\end{array}\right)$

$\text{Symbols for Coordinate Vectors}$

A special symbol is utilized to concisely convey the basis relative to which a coordinate vector is defined.

This notation is illustrated here using the vector space${\mathbb{R}}^3$and two of its bases.

Let$u$and$v$be two bases in${\mathbb{R}}^3$.

The basis$u$consists of three vectors$u_0,u_1,u_2$;$$they are called basis vectors of$u$.

$u=\left(\begin{array}{lll}{u}_0,& u_1,& u_2\end{array}\right)$

The basis$v$also consists of three vectors$v_0,v_1,v_2$;$$they are called basis vectors of$v$.

$v=\left(\begin{array}{lll}{v}_0,& v_1,& v_2\end{array}\right)$

The basis vectors are linearly independent, and each one is a column of three real numbers.

$u_0=\left(\begin{array}{l}{x}_0\\ x_1\\ x_2\end{array}\right)u_1=\left(\begin{array}{l}{y}_0\\ y_1\\ y_2\end{array}\right)u_2=\left(\begin{array}{l}{z}_0\\ z_1\\ z_2\end{array}\right)$

$v_0=\left(\begin{array}{l}{x}_0\\ x_1\\ x_2\end{array}\right)v_1=\left(\begin{array}{l}{y}_0\\ y_1\\ y_2\end{array}\right)v_2=\left(\begin{array}{l}{z}_0\\ z_1\\ z_2\end{array}\right)$

For example, one possible$u$looks like this:

$u=\left(\begin{array}{lll}\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right),& \left(\begin{array}{c}0\\ 1\\ 0\end{array}\right),& \left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)\end{array}\right)$

This is the standard basis, the simplest one.

Any vector in vector space${\mathbb{R}}^3$can be described unambiguously$$relative to any given basis of${\mathbb{R}}^3$.$$This is done by taking the linear combination of the vectors in the given basis.

The set of scalars used in such a linear combination uniquely describes the vector with respect to the given basis.

Those scalars are called the coordinates (or coordinate vector) of the described vector relative to the given basis.

Let$\beta$be any vector in${\mathbb{R}}^3$.$$Therefore,$\beta$can be writen as the linear combination$$of the vectors in basis$u$:

$\beta =j_0{u}_0+j_1{u}_1+j_2{u}_2$

Let${\left[\beta \right]}_u$denote the$$coordinates (or coordinate vector) of$\beta$relative to the basis$u$.

From the linear combination expression above,${\left[\beta \right]}_u$can be written as:

${\left[\beta \right]}_u=\left(\begin{array}{l}{j}_0\\ j_1\\ j_2\end{array}\right)$

Furthermore,$\beta$can also be written as the linear combination$$of the vectors in the other basis$v$:

$\beta =k_0{v}_0+k_1{v}_1+k_2{v}_2$

Let${\left[\beta \right]}_v$denote the$$coordinates (or coordinate vector) of$\beta$relative to the basis$v$.

From the linear combination expression above,${\left[\beta \right]}_v$can be written as:

${\left[\beta \right]}_v=\left(\begin{array}{l}{k}_0\\ k_1\\ k_2\end{array}\right)$

The special symbol${\left[\beta \right]}_u$can be employed to concisely state that$\beta$is defined relative to the basis$u$.

Similarly, the special symbol${\left[\beta \right]}_v$can be used to concisely state that$\beta$is defined relative to the basis$v$.

---

[This content was created with P64, available on the App Store]