Prove that two homogeneous systems of linear equations are equivalent iff they have the same solution set.My definition of equivalent of linear equations is that each equation of system A is a linear combination of the equations of system B and converse.So i have a bit trouble proving this statement in general as m×n form. I worked with their coefficient matrices and if i show that these two are row equivalent the problem is solved. Is there any method to use for this? If not what's the general prove.

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Prove that two homogeneous systems of linear equations are equivalent iff they have the same solution set.My definition of equivalent of linear equations is that each equation of system A is a linear combination of the equations of system B and converse.So i have a bit trouble proving this statement in general as m×n form. I worked with their coefficient matrices and if i show that these two are row equivalent the problem is solved. Is there any method to use for this? If not what&#039;s the general prove.

marinacabellok6k · Accepted Answer

This cannot be done without the rank theorem or some other theorem of similar power.We are given an (mA×n)-matrix A and an (mB×n)-matrix B. Denote by {ker}(A)⊂{R}n the solution space of Ax=0, by Ai!−∈{R}n the ith row vector of A, and byrow(A)=⟨A1−,…,AmA−⟩⊂Rnthe row space of A. Similarly for B. We have to prove the following claim:row(A)=row(B)⇔ker(A)=ker(B)For this it is sufficent to proverow(A)⊂row(B)⇔ker(A)⊂ker(B)Proof of ⇒: Assume row(A)⊂row(B), and consider a vector x∈ker(A). Let Bi−&amp;gt;x=∑k=1nBikxk=0 be an equation of the B-system. As Bi−=∑j=1mAλjAj− for certain λj∈{R} we obtainBi−x=∑j=1mAλjAj−x=0Since this is true for all i∈[mB] it follows that x∈ker(B). (This was the easy part.)Proof of ⇐: If row(B)⧸⊂row(A) then there is a row Bi− of B that does not belong to row(A). Denote by A&#039; the matrix obtained from A by adding the row Bi− at the bottom. Then A&#039; has rank one larger than A, hence ker(A′)⊂ker(A) has dimension one less than ker(A). It follows that there are vectors x∈ker(A)ker(A′). These vectors do not satisfy the last A&#039;-equation Bi−x=0, hence do not belong to ker(B). This proves ker(A)⧸⊂ker(B).

Prove that two homogeneous systems of linear equations are equivalent iff they have the same solutio

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