- #1

- 241

- 46

- Homework Statement
- Find the vectors in the image of the matrix

- Relevant Equations
- -

Hello!

I have this system here $$ \left[ \begin{matrix} -2 & 4 & \\\ 1 & -2 & {} \end{matrix} \right]x +\begin{pmatrix} 2 \\\ y \end{pmatrix}u $$ Now although the problem is for my control theory class,the background is completely math(as is 90% of control theory)

Basically what I need to do is to find the image of this matrix,and than chose y so that the u vector is in the image of the matrix.

So since the vectors in the matrix are linearly dependent we can choose either. so $$ im(A) = \begin{pmatrix} -2 \\\ 1 \end{pmatrix} $$

Now I need to chose y. Now 1 would be the obvious choice but the actual problem is the -2. Since the 2 in the vector is already given to me I cannot change it. And here is my question; let's say I choose y to be -1 that would give me the vector (2 -1),that is not in the image but if I multiply that vector by -1 I get the image vector.So my question is am I allowed to do that? Can i take multiples of a vector for them to be still in my image? If not is it possible that there is no way to choose y for the vector to be in the image of the matrix?

Thanks!

I have this system here $$ \left[ \begin{matrix} -2 & 4 & \\\ 1 & -2 & {} \end{matrix} \right]x +\begin{pmatrix} 2 \\\ y \end{pmatrix}u $$ Now although the problem is for my control theory class,the background is completely math(as is 90% of control theory)

Basically what I need to do is to find the image of this matrix,and than chose y so that the u vector is in the image of the matrix.

So since the vectors in the matrix are linearly dependent we can choose either. so $$ im(A) = \begin{pmatrix} -2 \\\ 1 \end{pmatrix} $$

Now I need to chose y. Now 1 would be the obvious choice but the actual problem is the -2. Since the 2 in the vector is already given to me I cannot change it. And here is my question; let's say I choose y to be -1 that would give me the vector (2 -1),that is not in the image but if I multiply that vector by -1 I get the image vector.So my question is am I allowed to do that? Can i take multiples of a vector for them to be still in my image? If not is it possible that there is no way to choose y for the vector to be in the image of the matrix?

Thanks!

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