# If n*n matrix, can row space ever be equal to null space?

• math_maj0r
In summary, the conversation discusses the relationship between the nullspace and rowspace of a matrix, and how they are complementary subspaces in R^n. It is not possible for the nullspace to be equal to the rowspace, as the only way for this to occur is if the matrix is "empty", which is not a valid matrix.
math_maj0r
If n*n matrix, can row space ever be equal to null space?

P.S.: this is NOT a homework question. It's a general question to get the concepts straight in my head.

Suppose v is a vector that belongs to the row and null spaces of A, then:

$$v^T = a^{T}A$$

And:

$$Av=\bold 0$$

Therefore, $v^{T}v$ must be equal to what? And what does this imply regarding v?

JSuarez said:
Suppose v is a vector that belongs to the row and null spaces of A, then:

$$v^T = a^{T}A$$

And:

$$Av=\bold 0$$

Therefore, $v^{T}v$ must be equal to what? And what does this imply regarding v?

What does this symbol mean? a^T

The transpose, of course.

The nullspace and rowspace of A are complementary subspaces in R^n (i.e. dim of nullspace + dim of rowspace = n and the dot of any vector in the nullspace with any vector in the rowspace will give 0)

vectors in the basis of the nullspace are of the form Ax=0

vectors in the basis of the rowspace are the pivots rows of the row reduced echelon form of A
Consider the zero Matrix

Noxide said:
The nullspace and rowspace of A are complementary subspaces in R^n (i.e. dim of nullspace + dim of rowspace = n and the dot of any vector in the nullspace with any vector in the rowspace will give 0)

vectors in the basis of the nullspace are of the form Ax=0

vectors in the basis of the rowspace are the pivots rows of the row reduced echelon form of A

Consider the zero Matrix

Consider the zero Matrix: it's nullspace will span R^n and it's Rowspace will be the zero vector.

Any invertible matrix has a rowspace that spans R^n and it's nullspace will be the zero vector

Any singular matrix will have k vectors in its rowspace basis and z vectors in its nullspace basis such that z + k = n

since the only vector dotted with itself that gives zero is the zero vector then the nullspace and rowspace would have to both be equal to the zero vector to give zero, meaning that the matrix must be "empty" for that to be possible and an "empty" matrix isn't a matrix at all, so there exists no matrix such that nullspace = rowspace

## 1. Can a square matrix have equal row and null spaces?

Yes, it is possible for a square matrix to have equal row and null spaces. This occurs when the rows of the matrix are linearly dependent, meaning that one or more rows can be expressed as a linear combination of the other rows.

## 2. Is it possible for a non-square matrix to have equal row and null spaces?

No, a non-square matrix cannot have equal row and null spaces. This is because the dimensions of the row and null spaces are determined by the number of rows and columns in the matrix, and a non-square matrix will have different numbers of rows and columns.

## 3. What does it mean when a matrix has equal row and null spaces?

When a matrix has equal row and null spaces, it means that the rows of the matrix are linearly dependent and can be expressed as a linear combination of each other. This also means that the null space of the matrix is non-trivial, as it contains more than just the zero vector.

## 4. How can you determine if a matrix has equal row and null spaces?

To determine if a matrix has equal row and null spaces, you can use the rank-nullity theorem. This theorem states that the dimension of the row space plus the dimension of the null space equals the number of columns in the matrix. If the dimensions of the row and null spaces are equal, then the matrix has equal row and null spaces.

## 5. What is the relationship between the row space and null space of a matrix?

The row space and null space of a matrix are complementary subspaces. This means that any vector in the null space is orthogonal to any vector in the row space, and vice versa. Additionally, the dimensions of the row and null spaces add up to the number of columns in the matrix.

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