Dimension of column- and rowspace

In summary, the column-space and row-space of a matrix refer to the span of its columns and rows, respectively. The dimension of these spaces can be calculated by counting the number of linearly independent columns or rows in the matrix. The dimension of column-space and row-space can be different, and a dimension of zero indicates that there are no linearly independent columns or rows. The rank of a matrix is equal to the dimension of both its column-space and row-space, and it is also equal to the number of non-zero rows in its reduced row echelon form.
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Niles
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[SOLVED] Dimension of column- and rowspace

Homework Statement


My book doesn't answer this question clearly, but I have notived the following connection:

Does the dimension of the rowspace of a matrix equal the dimension of the columnspace of a matrix which equals the rank of the matrix?
 
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If the matrix has real elements, the row and column spaces have the same dimension. Furthermore, the rank of A equals the dimension of the row space of A.
 

1. What is the difference between column-space and row-space?

The column-space refers to the span of the columns of a matrix, while the row-space refers to the span of the rows of a matrix. In other words, the column-space is the set of all possible linear combinations of the columns, while the row-space is the set of all possible linear combinations of the rows.

2. How do you calculate the dimension of column-space and row-space?

The dimension of the column-space is equal to the number of linearly independent columns in the matrix. Similarly, the dimension of the row-space is equal to the number of linearly independent rows in the matrix.

3. Can the dimension of column-space and row-space be different?

Yes, the dimension of column-space and row-space can be different. This is because the number of linearly independent columns may not be the same as the number of linearly independent rows in a matrix.

4. What does it mean if the dimension of column-space or row-space is zero?

If the dimension of column-space or row-space is zero, it means that there are no linearly independent columns or rows in the matrix. This indicates that the matrix only has the zero vector as a possible linear combination of its columns or rows.

5. How are the column-space and row-space related to the rank of a matrix?

The rank of a matrix is equal to the dimension of both its column-space and row-space. This means that the number of linearly independent columns or rows in a matrix is equal to the number of non-zero rows in its reduced row echelon form, which is the definition of rank.

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