Suppose 𝑇 ∈ ℒ(𝑉, 𝑊) and 𝑣 џ , ..., 𝑣 ֕ is a list of vectors in 𝑉 such that 𝑇𝑣1 , ..., 𝑇𝑣m ֕ is a linearly independent list in 𝑊. Prove that 𝑣1 , ..., 𝑣m ֕ is linearly independent. solution Suppose 𝑐1 , ..., 𝑐n ֙ ∈ 𝐅 are such that 𝑐1 𝑣1 + ⋯ + 𝑐n 𝑣n ֙ = 0. Applying 𝑇 to both sides of the equation above, we have 𝑐1 𝑇v1 + ⋯ + 𝑐n ֙ 𝑇𝑣n ֙ = 0. Because 𝑇𝑣1 , ... , 𝑇𝑣n ֕ is linearly independent, this implies that 𝑐1 = ⋯ = 𝑐n = 0. Thus 𝑣1 , ..., 𝑣n ֕ is linearly independent. Edward Frenkel --- Commentary: $\textbf{Exercise 4.}$ Suppose $T \in \mathcal{L}(V, W)$ and $v_1, \ldots, v_m$ is a list of vectors in $V$ such that $Tv_1, \ldots, Tv_m$ is a linearly independent list in $W$. Prove that $v_1, \ldots, v_m$ is linearly independent. $\textbf{Solution 4.}$ Suppose $c_1, \ldots, c_m \in \mathbb{F}$ are such that $c_1v_1 + \cdots + c_mv_m = 0.$ Applying $T$ to both sides of the equation above, we have $c_1Tv_1 + \cdots + c_mTv_m = 0.$ Because $Tv_1, \ldots, Tv_m$ is linearly independent, this implies that $c_1 = \cdots = c_m = 0.$ Thus $v_1, \ldots, v_m$ is linearly independent. $\textit{Commentary:}$ This exercise proves that linear independence is preserved under linear maps. More specifically, it shows that if a linear map $T$ sends a list of vectors to a linearly independent list, then the original list must also be linearly independent. The proof is straightforward and relies on the fundamental properties of linear maps. We start by assuming that the original list $v_1, \ldots, v_m$ is linearly dependent, so there are scalars $c_1, \ldots, c_m$, not all zero, such that $c_1v_1 + \cdots + c_mv_m = 0$. Then, we apply $T$ to both sides of this equation. Because $T$ is linear, it preserves addition and scalar multiplication, so we get $c_1Tv_1 + \cdots + c_mTv_m = 0$. But this contradicts the assumption that $Tv_1, \ldots, Tv_m$ is linearly independent. Therefore, our original assumption must be false, and $v_1, \ldots, v_m$ must be linearly independent. This result is useful in many contexts. For example, it can be used to prove that certain sets are linearly independent by showing that they map to linearly independent sets under a well-chosen linear map. It also provides insight into the structure-preserving properties of linear maps. $\textit{Examples:}$ 1. Let $T : \mathcal{P}_3(\mathbb{R}) \to \mathbb{R}^4$ be defined by $T(a + bx + cx^2 + dx^3) = (a, b, c, d)$. If $p_1, p_2, p_3, p_4 \in \mathcal{P}_3(\mathbb{R})$ are such that $Tp_1, Tp_2, Tp_3, Tp_4$ is linearly independent in $\mathbb{R}^4$, then $p_1, p_2, p_3, p_4$ is linearly independent in $\mathcal{P}_3(\mathbb{R})$. 2. Let $T : \mathbb{C}^3 \to \mathbb{C}^3$ be defined by $T(z_1, z_2, z_3) = (z_1 + z_2, z_2 + z_3, z_3 + z_1)$. If $v_1, v_2 \in \mathbb{C}^3$ are such that $Tv_1, Tv_2$ is linearly independent in $\mathbb{C}^3$, then $v_1, v_2$ is linearly independent in $\mathbb{C}^3$. 3. Let $T : \mathcal{P}(\mathbb{F}_2) \to \mathbb{F}_2^3$ be defined by $Tp = (p(0), p(1), p'(0))$. If $p_1, p_2, p_3 \in \mathcal{P}(\mathbb{F}_2)$ are such that $Tp_1, Tp_2, Tp_3$ is linearly independent in $\mathbb{F}_2^3$, then $p_1, p_2, p_3$ is linearly independent in $\mathcal{P}(\mathbb{F}_2)$. These examples illustrate the preservation of linear independence under various linear maps, including maps from polynomial spaces to $\mathbb{R}^n$ or $\mathbb{F}_2^n$, and maps from $\mathbb{C}^n$ to itself. They show how the result can be used to deduce the linear independence of a set by examining its image under a linear map.