Detailed Examples: Finding the Image and Basis of a Linear Map

📚 Quick Study Guide

🔍 Definition of Linear Map: A function $T: V \rightarrow W$ between vector spaces $V$ and $W$ is a linear map if for all vectors $u, v \in V$ and scalar $c$, $T(u + v) = T(u) + T(v)$ and $T(cu) = cT(u)$.

💡 Finding the Image: The image (or range) of a linear map $T$ is the set of all possible outputs: $Im(T) = \{T(v) : v \in V\}$. To find a basis for the image, apply $T$ to a basis of $V$ and then find a linearly independent subset that spans the resulting set.

📝 Finding the Kernel (Null Space): The kernel of a linear map $T$ is the set of all vectors in $V$ that map to the zero vector in $W$: $Ker(T) = \{v \in V : T(v) = 0\}$. To find a basis for the kernel, solve the equation $T(v) = 0$.

📐 Rank-Nullity Theorem: For a linear map $T: V \rightarrow W$, $dim(V) = dim(Ker(T)) + dim(Im(T))$, where $dim(V)$ is the dimension of V, $dim(Ker(T))$ is the dimension of the kernel of T (nullity), and $dim(Im(T))$ is the dimension of the image of T (rank).

Practice Quiz

1. What is the image of the linear map $T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ defined by $T(x, y) = (x + y, x - y)$?
   1. \(\{(0, 0)\} \)
   2. A line through the origin.
   3. \(\{\(x, y\) : x, y \in \mathbb{R}\} \) (i.e., all of \(\mathbb{R}^2\)).
   4. \(\{\(x, 0\) : x \in \mathbb{R}\} \)
2. What is the kernel of the linear map $T: \mathbb{R}^2 \rightarrow \mathbb{R}$ defined by $T(x, y) = x - y$?
   1. \(\{(0, 0)\} \)
   2. \(\{\(x, x\) : x \in \mathbb{R}\} \)
   3. \(\{\(x, -x\) : x \in \mathbb{R}\} \)
   4. \(\{\(x, y\) : x, y \in \mathbb{R}\} \)
3. Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ be defined by $T(x, y, z) = (x + y, y - z)$. What is the dimension of the kernel of $T$?
   1. 0
   2. 1
   3. 2
   4. 3
4. If $T: V \rightarrow W$ is a linear map and $dim(V) = 5$ and $dim(Ker(T)) = 2$, what is the dimension of $Im(T)$?
   1. 2
   2. 3
   3. 5
   4. 7
5. Which of the following sets could be a basis for the image of a linear transformation $T: \mathbb{R}^3 \rightarrow \mathbb{R}^2$?
   1. \(\{(1, 0, 0), (0, 1, 0)\} \)
   2. \(\{(1, 0), (0, 1), (1, 1)\} \)
   3. \(\{(1, 0), (0, 1)\} \)
   4. \(\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\} \)
6. Suppose $T: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ is a linear map defined by $T(x, y) = (x, x + y, y)$. What is a basis for the image of $T$?
   1. \(\{(1, 0, 0), (0, 1, 0), (0, 0, 1)\} \)
   2. \(\{(1, 1, 0), (0, 1, 1)\} \)
   3. \(\{(1, 0, 1), (1, 1, 1)\} \)
   4. \(\{(0, 0, 0)\} \)
7. Let $T: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be a linear transformation such that $T(1, 0, 0) = (1, 1, 0)$, $T(0, 1, 0) = (0, 1, 1)$, and $T(0, 0, 1) = (1, 0, 1)$. What is the image of the vector $(1, 1, 1)$ under $T$?
   1. \(\{(1, 1, 1)\} \)
   2. \(\{(2, 2, 2)\} \)
   3. \(\{(1, 2, 1)\} \)
   4. \(\{(2, 1, 2)\} \)