Let X be a n xx 1 column vector, whose components are all 0 except for the ith component, and A is a m xx n matrix, what is AX ?

Corbin Bradford

Corbin Bradford

Answered question

2022-09-20

Let X be a n × 1 column vector, whose components are all 0 except for the ith component, and A is a m × n matrix, what is AX ?
X = ( 0 1 0 )  and  A = ( a 11 a 1 n a m 1 a m n )
I've tried several examples for X and A, and I've noticed that there is a relation between the position of the ith component of X (which is 1), and AX, but I can't like really make the generalization.

Answer & Explanation

Absexabbelpjl

Absexabbelpjl

Beginner2022-09-21Added 8 answers

In this question first you should understand that when we post multiply any column X of size n × 1 to matrix A of size m × n then how AX look like ? If
X = ( x 1 x i x n )  and  A = ( a 11 a 1 n a m 1 a m n )
Then A X = x 1 ( a 11 a m 1 ) + x 2 ( a 12 a m 2 ) + + x i ( a 1 i a m i ) + + x n ( a 1 i a m i ) = ( a 1 i a m i )
In our case ,
X = ( 0 1 0 )  and  A = ( a 11 a 1 n a m 1 a m n )
Then
A X = 0 ( a 11 a m 1 ) + 0 ( a 12 a m 2 ) + + 1 ( a 1 i a m i ) + + 0 ( a 1 i a m i ) = ( a 1 i a m i )
Topniveauh2

Topniveauh2

Beginner2022-09-22Added 2 answers

Only one nonzero term enters each inner product and you get
X = ( a 1 i a 2 i a n i )

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