Exercise · ZW-CHANGE-OF-BASIS-006 · ★★☆ Intermediate
Let \(B=\{(1,2),(0,1)\}\) and \(B'=\{(1,0),(1,1)\}\). Which matrix takes \(B\)-coordinates to \(B'\)-coordinates?
Choose one option.
Show answer
- D. \(\begin{pmatrix}-1&-1\\2&1\end{pmatrix}\)
Explanation
Write each vector of \(B\) in terms of \(B'\): \((1,2)=-1(1,0)+2(1,1)\) and \((0,1)=-1(1,0)+1(1,1)\). These coordinates are the columns. Equivalently \(Q=(B')^{-1}B\).
Review the concept: Change of basis
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.
\[[x]_{B'}=P_{B'\leftarrow B}\,[x]_B\]
More practice problems: change of basis
- ZW-CHANGE-OF-BASIS-001 Let B=\(1,1),(1,-1)\. If [x]_B=(2,3), what is x in the standard basis?
- ZW-CHANGE-OF-BASIS-002 Let B=\(1,0),(1,1)\. What are the coordinates [x]_B of x=(3,2)?
- ZW-CHANGE-OF-BASIS-003 Let B=\(2,1),(1,1)\. Which matrix converts B-coordinates into standard coordinates?
- ZW-CHANGE-OF-BASIS-004 With B=\(2,1),(1,1)\, what are the coordinates [x]_B of x=(3,2)?
- ZW-CHANGE-OF-BASIS-005 If [x]_B=(1,0,0), which vector is x?
- ZW-CHANGE-OF-BASIS-007 Let B=\(1,1,0),(0,1,1),(0,0,1)\. What are the coordinates of x=(1,2,3) with respect to B?
- ZW-CHANGE-OF-BASIS-008 Let T(x)=Ax with A=[1 2; 0 3] and B=\(1,0),(1,1)\. What is the matrix [T]_B?
- ZW-CHANGE-OF-BASIS-009 The matrix taking B-coordinates to B'-coordinates is 2I. What is the relation between the bases?