True or false? If `A` and `B` have the same RREF then `R(A)=R(B)` . Here `R(A)` means the row space of `A.`

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degeneratecircle | High School Teacher | (Level 2) Associate Educator

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This is true. It can be proved that the elementary row operations used in getting the rref of a matrix do not change the row space. Let the rows of the matrix be `r_1,r_2,...,r_n.`

Operation 1 - Switching Rows:

Clearly,

`span{r_1,...,r_j,...,r_k,...,r_n}``=span{r_1,...r_k,...r_j,...,r_n},` so switching rows can;t change the row space.

Operation 2 - Multiplying a row by a nonzero constant:

It is also easy to see that

`span{r_1,...,cr_k,...r_n}=span{r_1,...,r_k,...r_n},` so this operation doesn't change the row space.

Operation 3 - Adding a multiple of one row to another row:

Since `a(r_j+br_k)=ar_j+(ab)r_k,` replacing `r_j` with `r_j+br_k` doesn't change the row space.

This proves that the row space isn't changed when going to rref.

Sources:

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