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2025-11-04 12:43:23 -06:00
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content/Math429/Math429_L18.md
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@@ -39,13 +39,13 @@ $T$ is surjective $\iff range\ T=W\iff null\ T'=0\iff T'$ injective
Let $V,W$ be a finite dimensional vector space, $T\in \mathscr{L}(V,W)$
Then $M(T')=(M(T))^T$. Where the basis for $M(T)'$ are the dual basis to the ones for $M(T)$
Then $M(T')=(M(T))^\top$. Where the basis for $M(T)'$ are the dual basis to the ones for $M(T)$
#### Theorem 3.133
$col\ rank\ A=row\ rank\ A$
Proof: $col\ rank\ A=col\ rank\ (M(T))=dim\ range\ T=dim\ range\ T'=dim\ range\ T'=col\ rank\ (M(T'))=col\ rank\ (M(T)^T)=row\ rank\ (M(T))$
Proof: $col\ rank\ A=col\ rank\ (M(T))=dim\ range\ T=dim\ range\ T'=dim\ range\ T'=col\ rank\ (M(T'))=col\ rank\ (M(T)^\top)=row\ rank\ (M(T))$
## Chapter V Eigenvalue and Eigenvectors