If A is an n×n matrix and b≠0 in Rn, then consider the set of solutions to Ax=b. Select true or false for each statement. 1. This set is a subspace 2. This set is closed under scalar multiplications 3. The set contains the zero vector 4. This set is closed under vector addition

Check the explanation

Step-by-step explanation:

1. FALSE. If X and Y are 2 solutions to the equation AX = b, then A(X+Y) = AX +AY = b+b = 2b ≠ b as b ≠ 0. This implies that X+Y is not a solution to the equation AX = b. Hence the set of solutions to the equation AX = b is not closed under vector addition.

2. FALSE. If X is a solution to the equation AX = b and if k is an arbitrary scalar other than 1, then A(kX) = kAX = kb≠ b. This implies that kX is not a solution to the equation AX = b. Hence the set of solutions to the equation AX = b is not closed under scalar multiplication.

3. FALSE, unless b = 0. A.0 = 0 . Since b ≠ 0, the set of solutions to the equation AX = b does not contain the 0 vector.

4. FALSE. In view of 1,2,3 above, the set of solutions to the equation AX = b is not a subspace since b ≠ 0.

not only in mathematics, scientific proof also are required in the other field of science. also maths proofs are of particular structure, containing steps and logics. so proving is like doing science, it is science. that's why it is important to prove. 

an inaccurate proof leads to errors, or stating results that are not accurate.  which wil mess the probelm up you dont want this

m∠5 + m∠3 = m∠1   is true.

m∠5 + m∠6 = 180 is true

m∠2 +m∠3 = m∠6   is true

m∠2 +m∠3 +m∠5 = 180   is true