SOLUTION: Hi I have a question about Vector space My question is the reverse question of the question in the book (Problem 17) Find the condition that make the solution set o

Question 32995This question is from textbook elementary linear Algebra
: Hi I have a question about Vector space
My question is the reverse question of the question in the book (Problem 17)
Find the condition that make the solution set of consistent system Ax=b (m linear equations, n unknowns) form a subspace of Rn) This question is from textbook elementary linear Algebra

Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
consistent system Ax=b (m linear equations, n unknowns)
form a subspace of Rn?
Note b is in R^m, A an n x m matrix
Let V = { x in R^n| Ax = b }
Since any vector space must contain the zero vector, b must be 0.
Clearly if x,y in V and scalrs a,b , we have
A(ax+by) = a Ax + b Ay = 0.
Hence, V = { x in R^n| Ax = 0 } is a subspace of R^n.
This proves b = 0 is the sufficient (& necessary) condition for the solution
set V to be a subspace of R^n.
Kenny
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