Question

Suppose vectors v1 ,... vp span Rn, and let T: Rn -> Rn be a linear transformation. Suppose T(vi) = 0 for i =1, ..., p. Show that T is a zero transformation. That is, show that if x is any vector in Rn, then T(x) = 0.

Answer 1

If the vectors $\{\mathbf v_1,\mathbf v_2,\ldots,\mathbf v_p\}$ span $\mathbb R^n$, then any vector $\mathbf x\in\mathbb R^n$ can be expressed as a linear combination of these vectors. This means there exist scalars $\{c_1,c_2,\ldots,c_p\}$ such that

$\mathbf x = c_1\mathbf v_1 + c_2\mathbf v_2 + \cdots + c_p\mathbf v_p$

Since T is a linear transformation, we have for all x,

$T(\mathbf x) = T(c_1\mathbf v_1 + c_2\mathbf v_2 + \cdots + c_p\mathbf v_p) \\\\ T(\mathbf x) = c_1T(\mathbf v_1) + c_2T(\mathbf v_2) + \cdots + c_pT(\mathbf v_p) \\\\ T(\mathbf x) = \mathbf 0 + \mathbf 0 + \cdots + \mathbf 0 = \mathbf 0$