The Brainliest Answer!

This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.

  We have to find  a matrix A of size  3 X 3 ( for example ) which has the vector v = [1,2,3] in the row space as well as column space.

So choose v as the first row.  Next, write transpose of v in the 1st column.  Then write 2nd row as a multiple of 1st row.  Then write 3rd row so that it is not dependent on 1st row. After that perform row reduction operations to get a echelon form.

A= \left[\begin{array}{ccc}1&2&3\\2&4&6\\3&0&5\end{array}\right] =>\left[\begin{array}{ccc}1&2&3\\0&0&0\\3&0&5\end{array}\right]=\left[\begin{array}{ccc}0&2&4/3\\0&0&0\\3&0&5\end{array}\right]\\\\Row\ Space\ of\ A=\{ [1,2,3],[3,0,5]\},\ \ or\ \ \{[0,2,4/3 ], [3,0,5 ] \}\\\\Column\ 3\ in\ row\ reduced\ Echelon\ form=v3=v2*2/3+v1*5/3.\\\\Column\ space\ of\ A=\{ [1,2,3 ], [2,4,0 ] \}

Null space of A = [ 2,4,6]  as it is linearly dependent on 1st and 3rd rows.
2)  Find a matrix with v = [1,2,3]  in null space and column space.

Write v as the first column.  Write v as the first row as well.  But it has to be dependent on 2nd and/or third row, so that it becomes O during row reduction operation. write the last two elements in the matrix in third row and perform row reduction in such a way that a column other than 1 becomes dependent.

B= \left[\begin{array}{ccc}1&2&3\\2&4&6\\3&1&0\end{array}\right]=\left[\begin{array}{ccc}0&0&0\\2&4&6\\3&1&0\end{array}\right]=\left[\begin{array}{ccc}0&0&0\\0&10/3&6\\3&1&0\end{array}\right]\\\\Column2=v2=10/18*v3+3*v1\\\\Hence,\ column\ space=\{ [1,2,3 ]^T, [3,6,0 ]^T \}\\

Null space of B = { [1,2,3] }  

There are four subspaces.   Column space,  row space, null space and left null space.  Null space is the set of rows in A which are linearly dependent on others.  Left null space is the null space of Matrix A^T (transpose of A).

A vector  can not possibly be in Column space and left null space.
A vector can not be possibly in row space and null space.

Because these pairs are complementary to each other.


We can also write remaining two elements in matrix A as x and y, after writing v as first row and column and a multiple of v as 2nd row.  Then find their values so that one desired column or row becomes linearly dependent or independent. 

2 5 2