Question 1205688: If T: R³→ R³ is If is a linear transformation defined by T(x,y,z)=(x+2y,x−y+z,−2y+z) Write the primary decomposition of R³
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! 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.
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