Question 1205688
To find the primary decomposition of R³ under the linear transformation T, we need to find the eigenvalues and eigenvectors of the matrix associated with T. 

**Step 1: Find the Matrix Representation of T**

The matrix representation of T is:

```
A = [[1, 2, 0],
     [1, -1, 1],
     [0, -2, 1]]
```

**Step 2: Find the Eigenvalues of A**

To find the eigenvalues, we solve the characteristic equation det(A - λI) = 0:

```
det([[1-λ, 2, 0],
     [1, -1-λ, 1],
     [0, -2, 1-λ]]) = 0
```

Solving this equation, we find the eigenvalues: λ₁ = 1, λ₂ = -1, and λ₃ = 1.

**Step 3: Find the Eigenspaces**

For each eigenvalue, we find the corresponding eigenspace:

* **For λ₁ = 1:**
  Solve the equation (A - I)v = 0:
  ```
  [[0, 2, 0],
   [1, -2, 1],
   [0, -2, 0]]v = 0
  ```
  The eigenspace E₁ is spanned by {(1, 0, 1), (0, 1, 1)}.

* **For λ₂ = -1:**
  Solve the equation (A + I)v = 0:
  ```
  [[2, 2, 0],
   [1, 0, 1],
   [0, -2, 2]]v = 0
  ```
  The eigenspace E₂ is spanned by {(1, -1, 1)}.

**Step 4: Primary Decomposition**

Since the eigenvalues are distinct, the primary decomposition of R³ is the direct sum of the eigenspaces:

```
R³ = E₁ ⊕ E₂
```

where:
* E₁ = span{(1, 0, 1), (0, 1, 1)}
* E₂ = span{(1, -1, 1)}

This means that every vector in R³ can be uniquely expressed as a sum of a vector in E₁ and a vector in E₂.