Chapter R: Respresentations

Question C10, Page 512

To find a solution for:

\[\rho_B(y)\]

it is necessary to find scalars:

\[a_1, a_2, a_3\]

such that:

\[y = a_1u_1 + a_2u_2 + a_3u_3\]

We get the following system of equations:

2a + 1b + 3c = 11

-2a + 3b + 5c = 5

2a + 1b + 2c = 8

We can solve this using matrix row operations and substitution.

```
## [,1] [,2] [,3] [,4]
## [1,] 2 1 3 11
## [2,] -2 3 5 5
## [3,] 2 1 2 8
```

First, subtract row 1 from row 3.

```
## [,1] [,2] [,3] [,4]
## [1,] 2 1 3 11
## [2,] -2 3 5 5
## [3,] 0 0 -1 -3
```

From this step, we get that -1c = -3. Solving for c, we get **c = 3**. We can substitute this into the system of equations and create a new matrix:

2a + 1b + 3(3) = 11

-2a + 3b + 5(3) = 5

The equations above become this:

2a + 1b = 2

-2a + 3b = -10

Adding the two equations together, we get:

4b = -8

**b = -2**

Finally, solving for a, we get:

2a - 2 = 2

2a = 4

**a = 2**

Therefore, the representation is:

\[\rho_B(y) = \left(\begin{array}{cc} 2\\ -2\\ 3 \end{array}\right)\]: