Question

Let V = W = R2. Choose the basis B = {x1, x2} of V , where x1 = (2, 3), x2 = (4,−5) and choose the basis D = {y1,y2} of W, where y1 = (1,1), y2 = (−3,4). Find the matrix of the identity linear mapping I : V → W with respect to these bases.

Answer #1

Let A =

1 |
-3 |
2 |
4 |

1 |
4 |
3 |
-5 |

The RREF of A is

1 |
0 |
17/7 |
1/7 |

0 |
1 |
1/7 |
-9/7 |

Hence (2,3) = (17/7)(1,1)+(1/7)(-3,4) and (4,-5) = (1/7)(1,1)-(9/7)(-3,4).

Let I: V →W be the identity linear mapping. Then I(
x_{1}) = I(2,3) = (2,3) = (17/7)(1,1)+(1/7)(-3,4) and I (
x_{2}) = I(4,-5) = (4,-5)= (1/7)(1,1)-(9/7)(-3,4).

Thus, the matrix of the identity linear mapping I : V →W with respect to the bases B and D is M =

17/7 |
1/7 |

1/7 |
-9/7 |

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