Exercise · ZW-CHANGE-OF-BASIS-009 · ★★★ Advanced
The matrix taking \(B\)-coordinates to \(B'\)-coordinates is \(2I\). What is the relation between the bases ?
Choose one option.
A Each vector of \(B'\) is twice the corresponding vector of \(B\) B Each vector of \(B\) is twice the corresponding vector of \(B'\) C The bases are equal D \(B'\) has twice as many vectors as \(B\) Show answer B. Each vector of \(B\) is twice the corresponding vector of \(B'\)← Back to the question
Explanation Column \(j\) is \([b_j]_{B'}=2e_j\), that is, \(b_j=2b_j'\). This is natural: with shorter basis vectors (\(B'\)) the coordinates must be larger.
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.
\[[x]_{B'}=P_{B'\leftarrow B}\,[x]_B\]
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