# Do row operations change the column space?

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## What do row operations change?

You can do the other row operations that you’re used to, but they change the value of the determinant. The rules are: If you interchange (switch) two rows (or columns) of a matrix A to get B, then det(A) = –det(B).

## Why do row operations not change row space?

In summary, after doing any elementary operation to v1,…,vn, the span doesn’t change. It follows directly that if A and B are row equivalent, since the rows of B can be obtained by elementary operations from the rows of A, the spans of their rows are equal.

## How do row operations change determinants?

Computing a Determinant Using Row Operations

If two rows of a matrix are equal, the determinant is zero. If two rows of a matrix are interchanged, the determinant changes sign. If a multiple of a row is subtracted from another row, the value of the determinant is unchanged.

## Why is it called left null space?

If one understands the concept of a null space, the left null space is extremely easy to understand. The word “left” in this context stems from the fact that ATy=0 is equivalent to yTA=0 where y “acts” on A from the left.

## What is column space and null space?

The column space of the matrix in our example was a subspace of R4. The nullspace of A is a subspace of R3. … the nullspace N(A) consists of all multiples of 1 ; column 1 plus column -1 2 minus column 3 equals the zero vector. This nullspace is a line in R3.

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## What is the dimension of a left null space?

As the above equation, EA = R is the same as solving y’ A = 0, and from the matrix R we can see that rank(A) = 2. So, the dimension of left null space = 3-2 = 1, i.e. dimension of column space-rank.

## How do row operations affect the column space?

Elementary row operations affect the column space. So, generally, a matrix and its echelon form have different column spaces. However, since the row operations preserve the linear relations between columns, the columns of an echelon form and the original columns obey the same relations.

## Do row operations change span?

Elementary row operations do not alter the row space. Thus a matrix and its echelon form have the same row space. The pivot rows of an echelon form span the row space of the original matrix.