Constructing Matrix E and F for RowA and NulA Basis | Homework Explanation

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In summary, a matrix E can be constructed with its rows as the basis vectors for rowA and a matrix F can be constructed with its columns as the basis vectors for nulA. When computing EF, the resulting matrix is the zero matrix. This is because the null space of a matrix is the set of vectors that get mapped to the zero vector when multiplied by the matrix. Since the basis vectors for the null space were used in the construction of F, multiplying by them will result in the zero matrix.
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Homework Statement


Construct a matrix E such that its rows are the basis vectors for rowA and a matrix F such that its columns are the basis vectors for nulA. Compute EF. Explain your results.


Homework Equations


Basis for NulA was { [3 2 1 0], [1 3 0 1] } (except vertical)
Basis for RowA was { [1 0 2 4], [0 1 3 2] } (except vertical)


The Attempt at a Solution


I computed EF and I got the zero matrix, but I'm not sure exactly why this is the case. Can someone provide some insight on this? Thanks.
 
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What is nullspace?
 
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NoMoreExams said:
What is nullspace?
Edit: DOH, nevermind, I get it now. The null space is the vectors that get mapped to the zero vector by the matrix. So if you take the matrix's row space and multiply by each of the basis vectors of the nullspace, you will get zero vectors in return (or since it was a matrix of the basis vectors to the null space, you will get the zero matrix in return).
 
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1. How do you construct Matrix E and F for RowA and NulA Basis?

Matrix E and F for RowA and NulA Basis can be constructed by following these steps:

  • Step 1: Find the basis for RowA and NulA using the row reduction method.
  • Step 2: Write down the basis vectors in the columns of Matrix E and Matrix F respectively.
  • Step 3: Use the basis vectors to construct the respective matrices by writing them as linear combinations of the standard basis vectors.
  • Step 4: Apply row operations to transform the matrices into reduced row echelon form.

2. What is the purpose of constructing Matrix E and F for RowA and NulA Basis?

The purpose of constructing Matrix E and F for RowA and NulA Basis is to obtain a clear representation of the linear transformation between the two vector spaces. Matrix E represents the matrix of the linear transformation from the standard basis to the basis for RowA, while Matrix F represents the matrix of the linear transformation from the standard basis to the basis for NulA.

3. Can Matrix E and F be constructed for any vector space?

Yes, Matrix E and F can be constructed for any vector space as long as the basis vectors can be determined.

4. What is the significance of Matrix E and F being in reduced row echelon form?

Matrix E and F being in reduced row echelon form allows for easier computation of the linear transformation between the two vector spaces. It also provides a clear representation of the basis vectors and their relationships with the standard basis.

5. How does constructing Matrix E and F help in solving problems related to vector spaces?

Constructing Matrix E and F is essential in solving problems related to vector spaces as it allows for a more accurate and efficient computation of linear transformations and their properties. It also helps in determining the dimension, rank, and nullity of the vector spaces, which are important concepts in linear algebra.

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