Find basis of solutions of this linear system Supposed to find basis of the subspace of vector spac

lilmoore11p8 2022-07-07 Answered
Find basis of solutions of this linear system
Supposed to find basis of the subspace of vector space R 3 of solutions of this linear system of equations:
y = { x 1 + 2 x 2 x 3 = 0 2 x 1 + 7 x 2 2 x 3 = 0 x 1 + 3 x 2 + x 3 = 0
I solve this system and I got: x 1 = x 3 and x 2 = 0
x = [ x 1 0 x 1 ] = x 1 [ 1 0 1 ] + 0 [ 0 0 0 ]
Is the basis: [ 1 0 1 ] ?
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Answers (1)

thatuglygirlyu
Answered 2022-07-08 Author has 14 answers
The question was basically answered in comments, I will add an answer to remove it from the list of unanswered question.
By adding the first and the third equation we get 5 x 2 = 0 which implies x 2 = 0. If we now plug x 2 = 0 into remaining equations, we see, that they are all multiples of x 1 x 2 = 0.
Hence all solutions are of the form ( x 1 , 0 , x 1 ). I.e., the solutions form the subspace { ( x 1 , 0 , x 1 ) ; x 1 R }.
Every vector in this subspace is a multiple of ( 1 , 0 , 1 ), which means that vector ( 1 , 0 , 1 ) generates the subspace.
Hence this subspace is one-dimensional and basis consists of a single vector ( 1 , 0 , 1 ).
Of course, we could take any non-zero multiple of ( 1 , 0 , 1 ) instead. For example, vector ( 2 , 0 , 2 ) generates the same subspace.
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