How to find a non-zero vector in the column space of M

In summary, a non-zero vector $\begin{bmatrix}-12\\-8\\20\end{bmatrix}$ can be found in the column space of matrix $M$.
  • #1
shamieh
539
0
Let the matrix $M = \begin{bmatrix}-12&-12&16&-15\\-6&-8&-8&-10\\0&20&0&25\end{bmatrix}$

Find a non zero vector in the column space of $M$

Is it not true that $\begin{bmatrix}-12\\-8\\20\end{bmatrix}$ is a non zero vector in the column space of $M$ ? For some reason it keeps telling me "that is incorrect your answer doesn't seem to be a Vector"
 
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  • #2
shamieh said:
Let the matrix $M = \begin{bmatrix}-12&-12&16&-15\\-6&-8&-8&-10\\0&20&0&25\end{bmatrix}$

Find a non zero vector in the column space of $M$

Is it not true that $\begin{bmatrix}-12\\-8\\20\end{bmatrix}$ is a non zero vector in the column space of $M$ ? For some reason it keeps telling me "that is incorrect your answer doesn't seem to be a Vector"

Hi shameih,

It is true that $\begin{bmatrix}-12\\-8\\20\end{bmatrix}$ is a non-zero vector that is in the column space of $M$. I see nothing wrong with your answer. Is it possible that the way you inserted it into the computer might be incorrect?
 

FAQ: How to find a non-zero vector in the column space of M

1. What is a non-zero vector in the column space of M?

A non-zero vector in the column space of M is a vector that is a linear combination of the columns of matrix M. This means that the vector can be expressed as a sum of scalar multiples of the columns of M. The vector must also have at least one non-zero element to be considered non-zero.

2. Why is it important to find a non-zero vector in the column space of M?

Finding a non-zero vector in the column space of M is important because it helps us understand the relationships between the columns of M. It also allows us to determine if the columns of M are linearly independent, which is important in many applications of linear algebra.

3. How do I find a non-zero vector in the column space of M?

To find a non-zero vector in the column space of M, you can use the process of Gaussian elimination or row reduction to put the matrix in reduced row-echelon form. Then, the non-zero rows of the reduced matrix will correspond to the non-zero vectors in the column space of M.

4. Can there be more than one non-zero vector in the column space of M?

Yes, there can be multiple non-zero vectors in the column space of M. This is because the column space of M is a subspace of the vector space containing all linear combinations of the columns of M. Therefore, any vector that can be expressed as a linear combination of the columns of M will also be in the column space of M.

5. How does finding a non-zero vector in the column space of M relate to the rank of M?

The rank of M is equal to the number of linearly independent columns in M. Therefore, if you are able to find a non-zero vector in the column space of M, this means that the columns of M are linearly independent and the rank of M is equal to the number of non-zero vectors you were able to find.

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