Suppose A is a 4 4 matrix and b is a vector in R4 with the

Suppose A is a 4 × 4 matrix and b is a vector in R⁴ with the property that Ax = b has a unique solution. Explain why the columns of A must span R⁴.

Solution

Solution:-

Given A is a 4*4 matrix , b is a vector belongs to R4 and

Ax=b has a unique solution

Which implies the system of equations are consistant .

Which implies Rank of A would be 4.

Which implies There is no non zero row\'s in A, afrer Reducing A into echelon form .

From that we can say that A is linearly indepedent set

Hence \" A \" becomes bisis for R4 ( each column of \" A \" concidered to be a basis element )

Therefore Columns of A must be span R4 .

Hence proof .