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

Definition of the Linear Transformation T
The linear transformation T:R^4 -> R^3 is defined as follows:

T(x,y,z,w) = [x-y, z, i]
[x^2, z-w, x]
[y^3z-3w]

Explanation

Dimension of the Range of a Linear Transformation
The dimension of the range of a linear transformation is also known as the rank of the transformation. It represents the maximum number of linearly independent vectors in the range of the transformation.

Steps to Determine the Dimension of the Range
To determine the dimension of the range of the linear transformation T, we need to find the maximum number of linearly independent vectors in the range of T.

Matrix Representation of T
To analyze the range of T, we can represent T in matrix form.

T(x,y,z,w) = [1, -1, 0, 0] [x]
[0, 0, 1, 0] * [y]
[0, 0, 0, 1] [z]
[w]

The matrix representation of T is a 3x4 matrix.

Reduced Row Echelon Form (RREF)
To find the maximum number of linearly independent vectors in the range of T, we need to find the RREF of the matrix representation of T.

Performing row operations, we can reduce the matrix to its RREF form:

[1, -1, 0, 0] [x] [x]
[0, 0, 1, 0] * [y] = [y]
[0, 0, 0, 1] [z] [z]

The RREF form of the matrix shows that the first, third, and fourth columns are pivot columns, while the second column is a free column.

Number of Linearly Independent Vectors
Since there are three pivot columns, there are three linearly independent vectors in the range of T.

Dimension of the Range
The dimension of the range of T is 3, as there are three linearly independent vectors in the range.

Conclusion
The dimension of the range of the linear transformation T:R^4 -> R^3 is 3, as there are three linearly independent vectors in the range.