大师作文网线性代数证明题（英文）求高手!
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线性代数证明题（英文）求高手!
Let A be a 3x4 matrix and b1,b2 be two vectors in R3 such that both matrix equations Ax=b1 and Ay=b2 are consistent.Prove that there exists a vector R4 such that the set of solutions y to the second equation is the set of all vectors of the form x+p,where x is any solution of the first equation.Write the solutions in full,cite every theorem or fact that you are using.
Let A be a 3x4 matrix and b1,b2 be two vectors in R3 such that both matrix equations Ax=b1 and Ay=b2 are consistent.Prove that there exists a vector R4 such that the set of solutions y to the second equation is the set of all vectors of the form x+p,where x is any solution of the first equation.Write the solutions in full,cite every theorem or fact that you are using.
既然两个方程都有解,可以各取一解x1, x2, 令p=x2-x1, 那么Ap=b2-b1. 如果一个向量形如x+p, A(x+p)=Ax+Ap=b1+b2-b1=b2, 即它是第二个方程的解；反之,如果y是第二个方程的解,那么A(y-p)=Ay-Ap=b2-(b2-b1)=b1, 因此y=(y-p)+p, 其中y-p是第一个方程的解.