Brahmagupta's Algebra, with Arithmetic and mensuration,: From the Sanscrit PDF

By Brahmagupta

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Additional resources for Algebra, with Arithmetic and mensuration,: From the Sanscrit of Brahmegupta and Bháscara

Example text

Bm } di W . Indichiamo con AB(V ) la B(W ) matrice rappresentativa di T rispetto alla prima coppia di basi, e con AB(V ) la matrice rappresentativa di T rispetto alla seconda coppia di basi. Allora vale la relazione 36 2 Matrici B(W ) B(W ) AB(V ) = E AB(V ) E −1 , dove E e` la matrice rappresentativa del cambiamento di base E : V → V , e E e` la matrice rappresentativa del cambiamento di base E : W → W . 1. Sia V uno spazio vettoriale di dimensione finita n, e sia T : V → V un’applicazione lineare.

Xn sono numeri reali tali che x1 A1 + x2 A2 + . . + xn An = b, allora, per ogni j = 1, . . , n, abbiamo: xj = det(A1 , . . , A j−1 , b, A j+1 , . . , An ) , det A dove b occupa la j–esima colonna invece di A j . Dimostrazione. Per ogni indice i, dalle propriet`a del determinante segue che   a11 . . a1i xi . . a1n   xi det A = det  ... ... ...  an1 . . ani xi . . ann Ma sappiamo che se sommiamo ad una colonna un multiplo di una differente colonna, il determinante non cambia valore.

Am1 am2 . . amn vn dove [ai j ] e` la matrice associata all’applicazione T . Per questo motivo, d’ora in avanti, i vettori saranno scritti in colonna. 8. Molti si domandano perch´e il numero di righe della matrice associata a T sia la dimensione dello spazio di arrivo, e non di quello di partenza. In simboli, T : V → W , dove n = dimV e m = dimW , e` rappresentata da una matrice m × n, le cui colonne sono i coefficienti di sviluppo lungo la base di W dei singoli vettori della base di V .

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Algebra, with Arithmetic and mensuration,: From the Sanscrit of Brahmegupta and Bháscara by Brahmagupta


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