AcademyVector Spaces
Academy
Coordinates
Level 1 - Math I (Physics) topic page in Vector Spaces.
Principle
Coordinates are the scalar coefficients used to write a vector in a chosen ordered basis. The vector is the same geometric or physical object, but its coordinate list depends on the basis used to describe it.
In physics, changing coordinates can simplify a problem without changing the underlying displacement, velocity, state, or field. The basis changes the description, not the object being described.
Notation
\(B=(\mathbf b_1,\ldots,\mathbf b_n)\)
an ordered basis
\(\mathbf v\)
the vector being represented
\([\mathbf v]_B\)
the coordinate vector of \mathbf v relative to B
\(c_i\)
the coefficient multiplying the basis vector \mathbf b_i
\(standard basis\)
the usual coordinate-axis basis in \mathbb R^n
The ordered basis \(B\) must be named before the coordinate vector \([\mathbf v]_B\) has a definite meaning.
Method
- State the ordered basis \(B=(\mathbf b_1,\ldots,\mathbf b_n)\).
- Write \(\mathbf v=c_1\mathbf b_1+\cdots+c_n\mathbf b_n\).
- Convert the vector equation into scalar equations.
- Solve for the coefficients \(c_1,\ldots,c_n\).
- Write the coordinate vector in the same order as the basis: \([\mathbf v]_B=(c_1,\ldots,c_n)\).
- Check the answer by reconstructing \(\mathbf v\) from the coefficients and basis vectors.
Rules
Coordinate definition
\[\mathbf v=c_1\mathbf b_1+\cdots+c_n\mathbf b_n\quad\Rightarrow\quad [\mathbf v]_B=(c_1,\ldots,c_n)\]
Coordinate example
\[c_1(1,1)+c_2(1,-1)=(3,1)\]
Coordinate solution
\[c_1+c_2=3,\quad c_1-c_2=1\quad\Rightarrow\quad c_1=2,\ c_2=1\]
- Coordinates are unique once a basis is fixed.
- Basis order matters: swapping basis vectors swaps coordinate positions.
- Coordinate vectors depend on the chosen basis.
- Standard coordinates are only one special coordinate system.
Examples
Question
Find the coordinates of
\[\mathbf v=(3,1)\]
in the basis \[B=((1,1),(1,-1))\]
Answer
Solve
\[c_1(1,1)+c_2(1,-1)=(3,1)\]
Comparing components gives \[c_1+c_2=3\]
and \[c_1-c_2=1\]
Adding the equations gives \[2c_1=4\]
so \[c_1=2\]
Then \[2+c_2=3\]
so \[c_2=1\]
Therefore \[[\mathbf v]_B=(2,1)\]
Checks
- Name the basis before writing coordinates.
- Preserve the order of the basis vectors.
- Solve for coefficients; do not just copy standard components into a new basis.
- Reconstruct the vector to check the coordinate list.