Can a non square matrix span?

In summary, a non square matrix can span all of its column space as long as its columns are linearly independent. To determine if a non square matrix can span its row space, the matrix must have linearly independent rows, which can be found by calculating its rank. The main difference between a square matrix and a non square matrix in terms of spanning is that a square matrix can span both its row and column space, while a non square matrix may only be able to span one or the other. However, a non square matrix can still span a subspace of its row space. It is also possible for a non square matrix to span a space of higher dimensions by having linearly dependent columns.
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Can a non square matrix span?
 
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The vectors represented by rows or columns of any matrix can 'span' (though,probably, not the entire space).
 

1. Can a non square matrix span all of its column space?

Yes, a non square matrix can span all of its column space. The column space of a matrix is the set of all possible linear combinations of its columns. As long as the columns are linearly independent, the matrix can span its entire column space regardless of its dimensions.

2. How can I determine if a non square matrix can span its row space?

In order for a non square matrix to span its row space, it must have linearly independent rows. This can be determined by finding the rank of the matrix. If the rank is equal to the number of rows, then the matrix can span its row space.

3. What is the difference between a square matrix and a non square matrix in terms of spanning?

A square matrix can span both its row and column space, while a non square matrix may only be able to span one or the other. Additionally, a square matrix can have a determinant of 0, indicating that it does not span its entire space, while a non square matrix must have a determinant of 0 in order to not span its entire space.

4. Can a non square matrix span a subspace of its row space?

Yes, a non square matrix can span a subspace of its row space. This means that the matrix can span a smaller subset of its row space without spanning the entire space. This can happen when the rows of the matrix are linearly dependent and therefore do not span the entire space.

5. Is it possible for a non square matrix to span a space of higher dimensions?

Yes, it is possible for a non square matrix to span a space of higher dimensions. This can happen when the columns of the matrix are linearly dependent, allowing them to span a higher dimensional space. However, the matrix will still only have the same number of linearly independent columns as its original dimensions.

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