Question about row space basis and Column space basis

In summary, the conversation discusses the span of a subspace S of R^3 and how it can be found by putting the vectors into a matrix and reducing it to rref. The basis for the row space is <(1,-2,-5),(0,1,3)> and it is stated that this basis also spans S. The question is then raised about whether the basis for the column space also spans S, and a link is provided for further information on the relationship between the two bases.
  • #1
PsychonautQQ
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Say a subspace S of R^3 is spanned by a basis = <(-1,2,5),(3,0,3),(5,1,8)>

By putting these vectors into a matrix and reducing it to rref, a basis for the row space can be found as <(1,-2,-5),(0,1,3)>. Furthermore, the book goes on to say that this basis spans the subspace S.

Cool, not suprising.

My question then is if the basis for the column space also spans S. If so, that means span(basis of column space) = span(basis of row space)?

Why doesn't my book say this straight up!?
 
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  • #2
PsychonautQQ said:
Why doesn't my book say this straight up!?

It isn't clear what you mean by "this". Are you asking a question about the one particular problem or are you asking about a general statement that applies to all matrices? If it's a general statement, can you state the conjecture in mathematical form? (If ... such-as-such then ...so-and so.)

Not all matrices are square. The span of a set of row vectors might not be in the same vector space as the span of a set of column vectors.
 

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

The row space basis of a matrix refers to the set of linearly independent rows that span the vector space of the matrix. The column space basis, on the other hand, refers to the set of linearly independent columns that span the same vector space. In other words, the row space represents the space of all possible row combinations, while the column space represents the space of all possible column combinations.

2. Why are row space basis and column space basis important?

Row and column space bases are important because they help us understand the structure and properties of a matrix. They can also be used to solve systems of linear equations, find the rank of a matrix, and determine the dimension of the vector space.

3. How do you find the row space basis and column space basis of a matrix?

To find the row space basis, you can use row reduction operations to transform the matrix into reduced row echelon form. The nonzero rows of the resulting matrix will form the row space basis. For the column space basis, you can use the columns that correspond to the pivot columns in the reduced row echelon form of the matrix.

4. Can a matrix have more than one row space basis or column space basis?

No, a matrix can only have one row space basis and one column space basis. However, the basis vectors may not be unique, meaning there may be more than one set of vectors that can form a basis for the row or column space of a matrix.

5. How does the row space basis and column space basis relate to the rank of a matrix?

The rank of a matrix is equal to the number of linearly independent rows or columns in the matrix. This means that the number of vectors in the row space basis is equal to the number of vectors in the column space basis, and both are equal to the rank of the matrix. In other words, the dimension of the row and column spaces is equal to the rank of the matrix.

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