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Bowei Tang
Bowei Tang
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Given that $A = LU$, where $L =$ \begin{bmatrix}1&0&0\\2&1&0\\3&0&1\end{bmatrix} and $U =$ \begin{bmatrix}1&1&0&1\\0&0&1&1\\0&0&0&0\end{bmatrix}

I need to find the following without extensive calculation to show insight into the topic:

• the rowspace of A$A$.

• the columnspace of A$A$.

• the nullspace of A$A$.

• the nullspace of A^T$A^T$.

I know that the rowspace is just the non-zero rows from U. The columnspace is the columns of L$L$ that correspond to the pivot columns of U$U$. The nulspacenullspace is the nullspace of U$U$. But I can't find an efficient way to calculate a basis for $\text{null}(A^T)$. I know that $\text{null}(A^T)$ is the orthogonal projection of $\text{col}(A)$, but I still can't manage to find a short, easy solution based on some properties without calculating $A^T$ and doing it that way.

If anyone knows some properties to solve this, please let me know :)

Given that $A = LU$, where $L =$ \begin{bmatrix}1&0&0\\2&1&0\\3&0&1\end{bmatrix} and $U =$ \begin{bmatrix}1&1&0&1\\0&0&1&1\\0&0&0&0\end{bmatrix}

I need to find the following without extensive calculation to show insight into the topic:

• the rowspace of A.

• the columnspace of A.

• the nullspace of A.

• the nullspace of A^T.

I know that the rowspace is just the non-zero rows from U. The columnspace is the columns of L that correspond to the pivot columns of U. The nulspace is the nullspace of U. But I can't find an efficient way to calculate a basis for $\text{null}(A^T)$. I know that $\text{null}(A^T)$ is the orthogonal projection of $\text{col}(A)$, but I still can't manage to find a short, easy solution based on some properties without calculating $A^T$ and doing it that way.

If anyone knows some properties to solve this, please let me know :)

Given that $A = LU$, where $L =$ \begin{bmatrix}1&0&0\\2&1&0\\3&0&1\end{bmatrix} and $U =$ \begin{bmatrix}1&1&0&1\\0&0&1&1\\0&0&0&0\end{bmatrix}

I need to find the following without extensive calculation to show insight into the topic:

• the rowspace of $A$.

• the columnspace of $A$.

• the nullspace of $A$.

• the nullspace of $A^T$.

I know that the rowspace is just the non-zero rows from U. The columnspace is the columns of $L$ that correspond to the pivot columns of $U$. The nullspace is the nullspace of $U$. But I can't find an efficient way to calculate a basis for $\text{null}(A^T)$. I know that $\text{null}(A^T)$ is the orthogonal projection of $\text{col}(A)$, but I still can't manage to find a short, easy solution based on some properties without calculating $A^T$ and doing it that way.

If anyone knows some properties to solve this, please let me know :)

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Bowei Tang
Bowei Tang
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Efficient way to calculate basis of Nul$\text{Null}(A^T)$ from given LU$LU$ factorisation without calculating A$A$.

givenGiven that

A = L.U
$A = LU$, where L =$L =$ \begin{bmatrix}1&0&0\\2&1&0\\3&0&1\end{bmatrix} and U =$U =$ \begin{bmatrix}1&1&0&1\\0&0&1&1\\0&0&0&0\end{bmatrix}

I need to find the following without extensive calculation to show insight into the topic:

• the rowspace of A.

• the columnspace of A.

• the nulspacenullspace of A.

• the nulspacenullspace of A^T.

I know that the rowspace is just the non zero-zero rows from U. The columnspace is the columns of L that correspond to the pivot columns of U. The nulspace is the nullspace of U. But I cantcan't find aan efficient way to calculate a basis for nul(A^T)$\text{null}(A^T)$. I know that nul(A^T)$\text{null}(A^T)$ is the orthagonalorthogonal projection of col(A)$\text{col}(A)$, but I still cantcan't manage to find a short, easy solution based on some properties without calculating A^T$A^T$ and doing it that way.

If anyone knows some properties to solve this, please let me know :)

Efficient way to calculate basis of Nul(A^T) from given LU factorisation without calculating A

given that

A = L.U
L = \begin{bmatrix}1&0&0\\2&1&0\\3&0&1\end{bmatrix} and U = \begin{bmatrix}1&1&0&1\\0&0&1&1\\0&0&0&0\end{bmatrix}

I need to find the following without extensive calculation to show insight into the topic:

• the rowspace of A.

• the columnspace of A.

• the nulspace of A.

• the nulspace of A^T.

I know that the rowspace is just the non zero rows from U. The columnspace is the columns of L that correspond to the pivot columns of U. The nulspace is the nullspace of U. But I cant find a efficient way to calculate basis for nul(A^T). I know that nul(A^T) is the orthagonal projection of col(A), but I still cant manage to find a short easy solution based on some properties without calculating A^T and doing it that way.

If anyone knows some properties to solve this please let me know :)

Efficient way to calculate basis of $\text{Null}(A^T)$ from given $LU$ factorisation without calculating $A$.

Given that $A = LU$, where $L =$ \begin{bmatrix}1&0&0\\2&1&0\\3&0&1\end{bmatrix} and $U =$ \begin{bmatrix}1&1&0&1\\0&0&1&1\\0&0&0&0\end{bmatrix}

I need to find the following without extensive calculation to show insight into the topic:

• the rowspace of A.

• the columnspace of A.

• the nullspace of A.

• the nullspace of A^T.

I know that the rowspace is just the non-zero rows from U. The columnspace is the columns of L that correspond to the pivot columns of U. The nulspace is the nullspace of U. But I can't find an efficient way to calculate a basis for $\text{null}(A^T)$. I know that $\text{null}(A^T)$ is the orthogonal projection of $\text{col}(A)$, but I still can't manage to find a short, easy solution based on some properties without calculating $A^T$ and doing it that way.

If anyone knows some properties to solve this, please let me know :)

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Ruben g.
Ruben g.
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Efficient way to calculate basis of Nul(A^T) from given LU factorisation without calculating A

given that

A = L.U
L = \begin{bmatrix}1&0&0\\2&1&0\\3&0&1\end{bmatrix} and U = \begin{bmatrix}1&1&0&1\\0&0&1&1\\0&0&0&0\end{bmatrix}

I need to find the following without extensive calculation to show insight into the topic:

• the rowspace of A.

• the columnspace of A.

• the nulspace of A.

• the nulspace of A^T.

I know that the rowspace is just the non zero rows from U. The columnspace is the columns of L that correspond to the pivot columns of U. The nulspace is the nullspace of U. But I cant find a efficient way to calculate basis for nul(A^T). I know that nul(A^T) is the orthagonal projection of col(A), but I still cant manage to find a short easy solution based on some properties without calculating A^T and doing it that way.

If anyone knows some properties to solve this please let me know :)