Linear Transformation T: R³ -> R³

The given linear transformation T: R³ -> R³ is defined by the equation Tx,y,z = (x + 2y - z, y + z, x - 2z). To find the dimension of the kernel of T, we need to determine the number of linearly independent vectors that map to the zero vector in the codomain.

Kernel of a Linear Transformation

The kernel of a linear transformation T, also known as the null space, is the set of all vectors in the domain that map to the zero vector in the codomain. Mathematically, it can be represented as ker(T) = {v in R³ | Tv = 0}.

Finding the Kernel of T

To find the kernel of T, we need to solve the equation Tx,y,z = (0, 0, 0). Let's set up the system of equations:

x + 2y - z = 0 (Equation 1)
y + z = 0 (Equation 2)
x - 2z = 0 (Equation 3)

Solving the System of Equations

We can solve this system of equations using various methods such as substitution or elimination. Let's use elimination to solve the system:

Adding Equation 2 and Equation 3, we get:
y + z + x - 2z = 0
x - z = 0
x = z (Equation 4)

Substituting Equation 4 into Equation 1, we have:
x + 2y - z = 0
z + 2y - z = 0
2y = 0
y = 0 (Equation 5)

Substituting Equation 5 and Equation 4 back into Equation 2, we get:
y + z = 0
0 + z = 0
z = 0

Therefore, the system of equations has a unique solution: x = y = z = 0.

Kernel Dimension

Since the only solution to the system of equations is the trivial solution (x = y = z = 0), we can conclude that the kernel of T contains only the zero vector. Therefore, the dimension of the kernel of T is 0.

Conclusion

The dimension of the kernel of the linear transformation T: R³ -> R³ defined by Tx,y,z = (x + 2y - z, y + z, x - 2z) is 0. This means that the kernel only contains the zero vector, indicating that T is an injective (one-to-one) linear transformation.