What is dim range T is in linear algebra . Give an example

In linear algebra, the **dimension** of the range (also known as the **rank**) of a linear transformation \( T: V \to W \) is the number of linearly independent vectors in the image of \( T \). The range (or image) of \( T \), denoted as \( \text{Range}(T) \) or \( \text{Im}(T) \), is the set of all possible outputs \( T(v) \) for \( v \) in the domain \( V \).

The dimension of the range \( T \) (or the rank of \( T \)) is an important concept because it tells us how many dimensions the image of \( T \) spans in the codomain \( W \). It provides information about the "effectiveness" of the transformation in mapping the input space to the output space.

### Example
Consider a linear transformation \( T: \mathbb{R}^3 \to \mathbb{R}^2 \) defined by the matrix:
\[
A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}.
\]

The transformation \( T \) maps a vector \( \vec{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \) in \( \mathbb{R}^3 \) to a vector in \( \mathbb{R}^2 \) as follows:
\[
T(\vec{x}) = A \vec{x} = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} x_1 + 2x_2 + 3x_3 \\ 4x_1 + 5x_2 + 6x_3 \end{bmatrix}.
\]

To find the dimension of the range of \( T \), we look at the rank of the matrix \( A \). The rank is the number of linearly independent rows (or columns) in \( A \).

For matrix \( A \):
\[
A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix},
\]
the row reduction (Gaussian elimination) gives:
\[
\begin{bmatrix} 1 & 2 & 3 \\ 0 & -3 & -6 \end{bmatrix}.
\]
Since there are 2 non-zero rows, the rank of \( A \) is 2. Thus, the dimension of the range of \( T \) is 2.

### Summary
- The range of \( T \) is spanned by two vectors in \( \mathbb{R}^2 \).
- The dimension of the range \( T \) (rank of \( T \)) is 2.

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### Related Questions
1. What is the null space of a linear transformation, and how is its dimension determined?
2. How does the rank of a matrix relate to its invertibility?
3. What is the relationship between the rank of a matrix and its row and column spaces?
4. How does the rank-nullity theorem relate the dimensions of the domain, range, and null space?
5. Can you give an example of a linear transformation with rank 1?

### Tip
Always remember that the rank of a matrix provides insight into the system's solutions: a full-rank matrix has a unique solution (if square and invertible), while a lower rank indicates dependencies among rows or columns.

Learn about the dimension of the range (rank) of a linear transformation T in linear algebra. This example uses a matrix transformation from R^3 to R^2 to explain how to calculate the rank and the dimension of the range. Suitable for undergraduate students studying linear algebra.

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