Page 31 - Matrices theory and applications

1. Elementary Theory
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the tensor product, whose block form is

b 1m C
b 11 C
···
.
.
.
.
 .


.
.
B ⊗ C = 
b n1 C
···
Show that (B, C) → B ⊗ C is a bilinear map and that its range
spans M nn ×mm (K). Is this map onto? b nm C 

(b) If p, p ∈ IN ∗ and D ∈ M m×p (K), E ∈ M m ×p (K), then

compute (B ⊗ C)(D ⊗ E).
(c) Show that for every bilinear form φ : M n×m (K)×M n ×m (K) →
K, there exists one and only one linear form
L : M nn ×mm (K) → K

such that L(B ⊗ C)= φ(B, C).