(LINALG) : Nullspace of transpose : N(A^T)

The book likely used alpha = 1, which is also a valid choice. So, in summary, the book and your method both give the same basis for N(A^T) and both are valid choices for finding the basis. The book may have chosen a different value for alpha, but it still leads to the same result.
  • #1
FrogPad
810
0
I'm not sure if I am making a mistake, or my book is wrong, or if both answers are correct. But, it is confusing me, and I would like to know why. We are asked to find the basis of the following subspaces on the matrix A.

Find: [tex] R(A^T),\,\,N(A),\,\,\,R(A),\,\,N(A^T)[/tex]

I'm having trouble finding N(A^T). Here is how I'm doing it.

[tex]A = \left[ \begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 \\
1 & 1 & 2 & 2
\end{array} \right]
[/tex]


thus:
[tex]
A^T = \left[ \begin{array}{cccc}
1 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 \\
0 & 1 & 1 & 2 \\
0 & 1 & 1 & 2
\end{array} \right]
[/tex]

so...

[tex]
rref(A^T) = \left[ \begin{array}{cccc}
1 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 \\
0 & 0 & 0 & 0
\end{array} \right]
[/tex]

then we are left with...

[tex]
\begin{array}{c}
x_1+\alpha = 0 \\
x_2 + \alpha = 0 \\
x_3 + \alpha = 0 \\
x_4 = \alpha
\end{array}
[/tex]

which gives:

[tex]
\alpha\left[
\begin{array}{c}
-1\\
-1\\
-1\\
1
\end{array} \right]
[/tex]

The book gives:
[tex] \left[
\begin{array}{c}
1 \\
1 \\
1 \\
-1
\end{array} \right]
[/tex]

as the basis for [tex]N(A^T) [/tex]

Is this the same? And why?

I mean [tex] \alpha [/tex] can be anything, so if [tex] \alpha = -1 [/tex] then I get the same answer as the book. So spanning the set with either "my" vector, or the books accomplishes the same thing. It's just confusing to me why the book would not follow the algorithm to get the answer. Thanks in advance.
 
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  • #2
As you noted, using alpha = -1 makes your answer the same. They are the same answer.
 
  • #3


Your approach to finding the nullspace of A^T is correct. The reason why the book's answer is different is because they have chosen to use a different parameter, let's call it \beta, instead of \alpha. So instead of setting x_4 = \alpha, they set x_4 = \beta. This leads to a different basis vector for the nullspace, but both approaches are correct and will span the same subspace. It is just a matter of personal preference which parameter to use.

Additionally, the book's answer is a more standard form for a basis vector in the nullspace, where the leading coefficient is 1. This is not necessary, but it is often preferred for simplicity and consistency. So while your answer is also correct, it may not be in the most standard form.

In summary, both your answer and the book's answer are correct and will span the same subspace. The only difference is in the choice of parameter and the standard form of the basis vector.
 

1. What is the nullspace of the transpose of a matrix?

The nullspace of the transpose of a matrix is the set of all vectors that, when multiplied by the transpose of the matrix, result in a zero vector. In other words, it is the set of all vectors that are mapped to the zero vector by the transpose of the matrix.

2. How is the nullspace of the transpose related to the nullspace of the original matrix?

The nullspace of the transpose is the orthogonal complement of the row space of the original matrix. This means that the nullspace of the transpose and the nullspace of the original matrix are orthogonal to each other and span the entire vector space.

3. Can the nullspace of the transpose be empty?

Yes, the nullspace of the transpose can be empty if the original matrix is full rank, meaning that it has no zero eigenvalues. In this case, the transpose of the matrix will have no nullspace.

4. How can the nullspace of the transpose be used in applications?

The nullspace of the transpose can be used in many applications, such as finding the solution to a system of linear equations, determining the rank of a matrix, and finding the basis of a vector space. It can also be used in data compression and image processing algorithms.

5. How can the nullspace of the transpose be calculated?

The nullspace of the transpose can be calculated by finding the nullspace of the original matrix and then taking the orthogonal complement of that set of vectors. This can be done using row reduction techniques or by using software such as MATLAB or Python's NumPy library.

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