Collection of Maths Problems

Matrix of the change of basis

Task number: 2566

• Variant 1

  \([id]_{AK}\), i.e. from the basis \(A\) to the canonical basis.

• Hint

  Let \(\mathbf A\) be the matrix with the columns formed of the vectors of \(A\), analogously for \(\mathbf B\) and \(B\), then \((\mathbf B|\mathbf A)\sim…\sim (\mathbf I,[id]_{AB})\).

• Result

  \([id]_{AK}=\mathbf A = \begin{pmatrix} 1 & 4 & 3 & 2 \\ 2 & 1 & 1 & 0 \\ 0 & 3 & 3 & 2 \\ 1 & 1 & 4 & 2 \\ \end{pmatrix}\).

• Variant 2

  \([id]_{KB}\), i.e. from the canonical basis to the basis \(B\).

• Result

  \([id]_{KB}=\mathbf B^{-1}= \begin{pmatrix} 2 & 4 & 0 & 1 \\ 3 & 3 & 2 & 0 \\ 0 & 3 & 3 & 0 \\ 4 & 1 & 2 & 3 \\ \end{pmatrix} \).

• Variant 3

  \([id]_{AB}\), i.e. from the basis \(A\) to the basis \(B\).

• Result

  \([id]_{AB}=\mathbf B^{-1}\mathbf A= \begin{pmatrix} 1 & 3 & 4 & 1 \\ 4 & 1 & 3 & 0 \\ 1 & 2 & 2 & 1 \\ 4 & 1 & 1 & 3 \\ \end{pmatrix} \).