I have fairly simple question regarding the following problem.
Describe the following subspace as the span of a system of generators : $$\textsf V = \{(x,y,z,t)\in \Bbb R^4 :\, x + y -z = 0 \textrm{ and } 2x - y + z = 0\}$$
Question : Can this be computed. I am a little thrown off for the fact that I am given three variables $x$, $y$ and $z$ and I am supposed to be in the fourth dimension.
Assuming that this can be computed I believe I can solve it by turning it to echelon form. Putting all the variables in terms of the other, factoring out the variables I used and then saying the span is equal to the linear combinations of ...
I also know that because we are going from a homogeneous system of equations to the description of a subspace we would be using orthogonal vectors outside of the subspace to describe the subspace.
Is this correct? Thank you for your time.
I suspect there is no $w$, since you said you're in dimension $4$.
You can indeed use row echelon form, or Gaussian elimination (which is the same, but from the linear system perspective).
Clearly, the matrix of your linear system has rank $2$, so you will find a set of solutions of dimension $2$.
Gaussian elimination will be extremely quick.
Edit. The rank of a matrix is the dimension of the subspace generated by the rows of the matrix. One may show it is also the dimension of the subspace generated by the columns of the matrix.
The rank theorem says that if we have a linear homogeneous system, the dimension of the set of solutions equals to the number of variables (that is the number of columns of the corresponding matrix) minus the rank of the matrix. (Alternatively, you can define the rank by the number of pivots obtained using Gaussian elimination).
This is useful if you want to guess in advance the dimension of the set of solutions (provided you can compute easily the rank of your matrix), but it is absolutely not necessary.