Linear transformations · Definition

Change of basis

linear algebralinear transformations

Before you start you need to know

The same vector is described by different numbers depending on the chosen basis; a change of basis translates between those descriptions and lets us pick the most convenient one for each problem.

Definition

The change of basis \(P\) is the passage from a vector's coordinates with respect to a basis \(B\) to its coordinates with respect to another basis \(B'\) of the same space.

\[[x]_{B'}=P_{B'\leftarrow B}\,[x]_B\]

If \(B=\{b_1,\dots,b_n\}\) is a basis, the coordinates \([x]_B=(c_1,\dots,c_n)\) are the coefficients in \(x=c_1b_1+\dots+c_nb_n\). When the columns of a matrix \(P\) are the vectors of \(B\) written in the standard basis, \(x=P[x]_B\) and, since \(P\) is invertible, \([x]_B=P^{-1}x\). A change of basis is therefore multiplication by an invertible matrix.

Between any two bases \(B\) and \(B'\), the matrix \(P_{B'\leftarrow B}\) has as its \(j\)-th column the coordinates \([b_j]_{B'}\). It equals \((P_{B'})^{-1}P_B\), and the matrix of the opposite change is its inverse. Example: for \(B=\{(2,1),(1,1)\}\), the vector \(x=(3,2)\) has \([x]_B=(1,1)\), since \((2,1)+(1,1)=(3,2)\).

A change of basis also transforms the matrices of linear maps. If \(T(x)=Ax\) in the standard basis, its matrix with respect to \(B\) is \([T]_B=P^{-1}AP\). The two matrices describe the same map and are called similar. A good choice of \(B\) can make \([T]_B\) diagonal, which is the central idea of diagonalization.

Typical mistakes: mixing up \(P\) and \(P^{-1}\) (check which basis is on each side), reordering the columns (the basis is ordered), and forgetting that the vector itself does not change: only its numerical description does.

Think about it

If \(P\) takes \(B\)-coordinates to standard coordinates, which matrix takes standard coordinates to \(B\)-coordinates?

The inverse \(P^{-1}\), which exists because the columns of \(P\) form a basis and are independent.

If two vectors of \(B\) are swapped, do the coordinates of a vector change?

Yes: the two corresponding coordinates are swapped, because a basis is ordered.

Exercises 10

  1. Let \(B=\{(1,1),(1,-1)\}\). If \([x]_B=(2,3)\), what is \(x\) in the standard basis?
  2. Let \(B=\{(1,0),(1,1)\}\). What are the coordinates \([x]_B\) of \(x=(3,2)\)?
  3. Let \(B=\{(2,1),(1,1)\}\). Which matrix converts \(B\)-coordinates into standard coordinates?
  4. With \(B=\{(2,1),(1,1)\}\), what are the coordinates \([x]_B\) of \(x=(3,2)\)?
  5. If \([x]_B=(1,0,0)\), which vector is \(x\)?
  6. Let \(B=\{(1,2),(0,1)\}\) and \(B'=\{(1,0),(1,1)\}\). Which matrix takes \(B\)-coordinates to \(B'\)-coordinates?
  7. Let \(B=\{(1,1,0),(0,1,1),(0,0,1)\}\). What are the coordinates of \(x=(1,2,3)\) with respect to \(B\)?
  8. Let \(T(x)=Ax\) with \(A=\begin{pmatrix}1&2\\0&3\end{pmatrix}\) and \(B=\{(1,0),(1,1)\}\). What is the matrix \([T]_B\)?
  9. The matrix taking \(B\)-coordinates to \(B'\)-coordinates is \(2I\). What is the relation between the bases?
  10. Let \(T(x)=Ax\) with \(A=\begin{pmatrix}3&1\\0&2\end{pmatrix}\) and \(B=\{(1,0),(1,-1)\}\). What is \([T]_B\)?

What comes next

References

  • Gilbert Strang, Introduction to Linear Algebra, 6th edition, Wellesley-Cambridge Press, 2023. §8.3
  • Sheldon Axler, Linear Algebra Done Right, 4th edition, Springer, 2024. §3.D
  • David C. Lay, Steven R. Lay, Judi J. McDonald, Linear Algebra and Its Applications, 6th edition, Pearson, 2021. §4.4 y §4.7