What is subspace?

The definition of this simple work about to follow, are linear subspaces in a Matrix. We require to let our equation equal zero, and we will be finding two different solutions in a column vector where \(V_2,V_3\) and we will ask if it's linearly independant.

Our equation should be of the form.

\(\begin{pmatrix} x \\ y \\ z \end{pmatrix}= (V_1,AV_2,BV_3)\)

Thus if we have:

\(x+2y+3z=6\)

then

\((x,y,z)=(1,1,1)\)

As you can see, in its elementary form, by making them equal, 1,1 and 1, we can see it is a solution to the value \(x+2y+3z=6\). of course, by making sure we keep to what we are wanting in the beginning, we set it to zero:

\(x+2y+3z=0\)

We require two solutions, and to check if it is linearly independant, thus we can have:

\((-2,1,0)\) and \((-3,0,1)\)

From here, we can clearly see they are not linearly independant - thus the general solution of the entire form is:

\(\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}+A\begin{pmatrix} -2 \\ 1 \\ 0 \end{pmatrix}+B\begin{pmatrix} -3 \\ 0 \\ 1 \end{pmatrix}\)

And this is a mathematical description of a linear subspace in a matrix.

 

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