The following question seems to me interesting. it gives solution space and required the correspondi

Alisa Durham

Alisa Durham

Answered question

2022-05-15

The following question seems to me interesting. it gives solution space and required the corresponding system of equation. The question is the following:
Consider the vectors in R 4 defined by
a 1 = ( 1 , 0 , 1 , 2 ), a 2 = ( 3 , 4 , 2 , 5 ), a 3 = ( 1 , 4 , 0 , 9 )
Find a system of homogeneous linear equations for which the space of solutions is exactly subspace of R 4 spanned by the three given vectors.
First i check the linear independence of the given vectors to see form of the generated space. But after determining i only obtained the result that the rank of the coefficient matrix of the corresponding homogeneous system of equations is 2. i obtained this result by rank-nullity theorem. But i can't go further. Please help.
Thanks in advance...

Answer & Explanation

garcialdariamcy4q

garcialdariamcy4q

Beginner2022-05-16Added 15 answers

use your vectors a 1 T , a 2 T , a 3 T as rows and make up a matrix
B = ( 1 0 1 2 3 4 2 5 1 4 0 9 ) . the null space of B will be the coefficient matrix we are looking for. solving B ( x y z w ) = ( 0 0 0 ) you will find A = ( 4 1 4 0 8 11 0 4 ) so that B A T = ( 0 0 0 0 0 0 ) which is the same as A B T is the zero matrix.
so your homogeneous system of equations are A B T
Amappyaccon22j7e

Amappyaccon22j7e

Beginner2022-05-17Added 3 answers

1. Find the normal vector, i.e. x such that a i x = 0 for all i = 1 , 2 , 3
2. Your equation is a x = 0.

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