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.