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Scorpion4ik [409]
3 years ago
12

Determine if the vector u is in the column space of matrix A and whether it is in the null space of A.

Mathematics
1 answer:
Musya8 [376]3 years ago
4 0

Answer: d) Not in Col in Nul A

Step-by-step explanation: The definition of <u>Column</u> <u>Space</u> of an <em>m x n</em> 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, A.x=u.

To solve, first write the augmented matrix of the system:

\left[\begin{array}{cccc}1&0&3&-4\\-2&-1&-4&-5\\3&-3&0&3\\-1&3&6&1\end{array}\right]

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:

\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]

Which gives a system with impossible solutions.

But if A.x=0, there would be a solution.

<u>Null</u> <u>Space</u> of an <em>m x n</em> matrix is a set of all solutions to A.x=0, so vector u is a null space of A, denoted by null (A)

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