Exercise · ZW-CHANGE-OF-BASIS-008 · ★★☆ Intermediate
Let \(T(x)=Ax\) with \(A=\begin{pmatrix}1&2\\0&3\end{pmatrix}\) and \(B=\{(1,0),(1,1)\}\). What is the matrix \([T]_B\)?
Choose one option.
Show answer
- C. \(\begin{pmatrix}1&0\\0&3\end{pmatrix}\)
Explanation
\([T]_B=P^{-1}AP\) with \(P=\begin{pmatrix}1&1\\0&1\end{pmatrix}\). Directly: \(T(1,0)=(1,0)=1\,b_1\) and \(T(1,1)=(3,3)=3\,b_2\), so \(b_1\) stays fixed and \(b_2\) is multiplied by \(3\).
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-006 Let B=\(1,2),(0,1)\ and B'=\(1,0),(1,1)\. Which matrix takes B-coordinates to B'-coordinates?
- 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-009 The matrix taking B-coordinates to B'-coordinates is 2I. What is the relation between the bases?