I was given the following definition and proposition.Definition: The set { x ∈ R n | A x = 0 } is the null set of A.Proposition: If p is a vector such that Ap=b, then { x ∈ R n | A x = b } = { y + p | y ∈ N S ( A ) }I am just so perplexed by the sudden introduction of notations and all of the null set. Can someone please explain this in simple terms and explain what the notations are?

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I was given the following definition and proposition.Definition: The set   {  x  ∈            R        n        |    A  x  =  0  } is the null set of A.Proposition: If p is a vector such that Ap=b, then   {  x  ∈            R        n        |    A  x  =  b  } =   {  y  +  p      |    y  ∈  N  S  (  A  )  }I am just so perplexed by the sudden introduction of notations and all of the null set. Can someone please explain this in simple terms and explain what the notations are?

Jaelyn Levine · Accepted Answer

I would guess that this is the first time that you have encountered set builder notation. The expression   {  x  ∈            R        n    ∣  A  x  =  0  } can be expressed in English as &quot;the set of x in             R        n   such that   A  x  =  0&quot;. Here,             R        n   refers to the set of all column-vectors that contain n real numbers.To put this another way, the null set of A is the set of solutions to the equation      (                                        a                          11                                                ⋯                                      a                          1              n                                                            ⋮                          ⋱                          ⋮                                                  a                          m              1                                                ⋯                                      a                          m              n                                            )        (                                        x            1                                                ⋮                                                  x            n                                )    =      (                            0                                      ⋮                                      0                      )    ,where we think of the column of values       x    1    ,  …  ,      x    n   as the single vector &quot;x&quot;

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