ThmDex – An index of mathematical definitions, results, and conjectures.

Result R1371 on D281: Bilinear map

Bilinear map is zero if either argument is zero

Formulation 0

Let $f : V_1 \times V_2 \to W$ be a D281: Bilinear map such that

(i)	$0_1, 0_2, 0_W$ are each the D737: Zero vector in $V_1, V_2, W$, respectively

Let $x = (x_1, x_2) \in V_1 \times V_2$ be a D1129: Vector such that

(i)	$x_1 = 0_1$ or $x_2 = 0_2$

Then \begin{equation} f(x) = f(x_1, x_2) = 0_W \end{equation}

Proofs

Proof 0

Let $f : V_1 \times V_2 \to W$ be a D281: Bilinear map such that

(i)	$0_1, 0_2, 0_W$ are each the D737: Zero vector in $V_1, V_2, W$, respectively

Let $x = (x_1, x_2) \in V_1 \times V_2$ be a D1129: Vector such that

(i)	$x_1 = 0_1$ or $x_2 = 0_2$

This result is a particular case of R1370: Multilinear map is zero if any argument is zero. $\square$