Answer :
Answer: d) Not in Col in Nul A
Step-by-step explanation: The definition of Column Space of an m x n matrix A is the set of all possible combinations of the columns of A. It is denoted by col A. To determine if a vector is a column space, solve the matrix equation:
A.x = b or, in this case, [tex]A.x=u[/tex].
To solve, first write the augmented matrix of the system:
[tex]\left[\begin{array}{cccc}1&0&3&-4\\-2&-1&-4&-5\\3&-3&0&3\\-1&3&6&1\end{array}\right][/tex]
Now, find the row-echelon form of matrix A:
1) Multiply 1st row by 2 and add 2nd row;
2) Multiply 1st row by -3 and add 3rd row;
3) MUltiply 1st row by 1 and add 4th row;
4) MUltiply 2nd row by -1;
5) Multiply 2nd row by 3 and add 3rd row;
6) Multiply 2nd row by -3 and add 4th row;
7) Divide 3rd row by -15;
8) Multiply 3rd row by -15 and add 4th row;
The echelon form matrix will be:
[tex]\left[\begin{array}{cccc}1&0&3&-4\\0&1&-2&13\\0&0&1&-\frac{51}{15}\\0&0&0&-13 \end{array}\right][/tex]
Which gives a system with impossible solutions.
But if [tex]A.x=0[/tex], there would be a solution.
Null Space of an m x n matrix is a set of all solutions to [tex]A.x=0[/tex], so vector u is a null space of A, denoted by null (A)