SOLUTION: Show that the inner product ⟨u, v⟩ = 5u1v1 − u1v2 − u2v1 + 10u2v2 is the inner product on R2 generated by the matrix A= | 2 1 | | -1 3 |

Algebra ->  College  -> Linear Algebra -> SOLUTION: Show that the inner product ⟨u, v⟩ = 5u1v1 − u1v2 − u2v1 + 10u2v2 is the inner product on R2 generated by the matrix A= | 2 1 | | -1 3 |      Log On


   

Question 1167650: Show that the inner product ⟨u, v⟩ = 5u1v1 − u1v2 − u2v1 + 10u2v2 is the inner product on R2 generated by the matrix A=
| 2 1 |
| -1 3 |
Answer by Resolver123(6) About Me  (Show Source):
You can put this solution on YOUR website!
We have to show that the inner product defined by ⟨𝑢,𝑣⟩ = %28Au%29%5ET%2A%28Av%29 produces the expression:
⟨𝑢,𝑣⟩ = 5u%5B1%5Dv%5B1%5D-u%5B1%5Dv%5B2%5D-u%5B2%5Dv%5B1%5D%2B10u%5B2%5Dv%5B2%5D.


Let u+=+%28matrix%282%2C1%2Cu%5B1%5D%2Cu%5B2%5D%29%29 and v+=+%28matrix%282%2C1%2Cv%5B1%5D%2Cv%5B2%5D%29%29.
These imply that and .
These give
The proof is complete.