Linear Transformation T: R^4 -> R^3

To find the dimension of the range of a linear transformation, we need to determine the number of linearly independent vectors in the image of the transformation. In other words, we need to find the maximum number of vectors that can be formed from the image vectors such that none of them can be expressed as a linear combination of the others.

Definition of Linear Transformation T

The linear transformation T is defined as T(x, y, z, w) =
|x - y z w|
|x^2 z - w|
|x y 3z - 3w|

Finding the Image of T

To find the image of T, we need to apply the transformation to all possible input vectors (x, y, z, w) in R^4 and observe the resulting output vectors in R^3.

Let's consider the standard basis vectors in R^4: e1 = (1, 0, 0, 0), e2 = (0, 1, 0, 0), e3 = (0, 0, 1, 0), and e4 = (0, 0, 0, 1).

Applying T to e1
T(e1) = |1 - 0 0 0|
|1^2 0 - 0|
|1 0 0|

= |1 0 0|
|1 0 0|
|1 0 0|

Applying T to e2
T(e2) = |0 - 1 0 0|
|0^2 0 - 0|
|0 1 0|

= |-1 0 0|
| 0 0 0|
| 0 1 0|

Applying T to e3
T(e3) = |0 - 0 0 0|
|0^2 0 - 0|
|0 0 3|

= |0 0 0|
|0 0 0|
|0 0 3|

Applying T to e4
T(e4) = |0 - 0 0 1|
|0^2 0 - 1|
|0 0 3 - 3|

= |0 0 0|
|-1 0 0|
|0 0 0|

Observations
From the above calculations, we can observe that the image of T consists of the following three vectors in R^3:
v1 = (1, 0, 0)
v2 = (-1, 0, 0)
v3 = (0, 0, 3)

Linear Independence
To determine if these vectors are linearly independent, we need to check if any one of them can be expressed as a linear combination of the others.

Linear