Task:For n∈N and W≤Fn, show that there exists a homogeneous system of linear equations whose solution space is W.My work:Since W≤Fn,k=dim⁡(W)≤dim⁡(Fn). Let's say that {w1,w2,…,wk} is a basis of W. Now, construct a matrix A (of size k×n) such that its rows are elements from the basis of W, stacked together. The row space of A is W, so the row space of its row-echelon form is W too. At this point, I'm stuck! I'm trying to come up with a homogeneous system with the help of A, though there may exist other easier ways of approaching this problem.Could someone show me the light?P.S. W≤Fn stands for W is a subspace of Fn.P.P.S. Isn't this equivalent to saying that W is the null-space of some matrix? Can we go ahead along these lines, and construct a matrix P such that Pw=0 for all w∈W?

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Task:For n∈N and W≤Fn, show that there exists a homogeneous system of linear equations whose solution space is W.My work:Since W≤Fn,k=dim⁡(W)≤dim⁡(Fn). Let&#039;s say that {w1,w2,…,wk} is a basis of W. Now, construct a matrix A (of size k×n) such that its rows are elements from the basis of W, stacked together. The row space of A is W, so the row space of its row-echelon form is W too. At this point, I&#039;m stuck! I&#039;m trying to come up with a homogeneous system with the help of A, though there may exist other easier ways of approaching this problem.Could someone show me the light?P.S. W≤Fn stands for W is a subspace of Fn.P.P.S. Isn&#039;t this equivalent to saying that W is the null-space of some matrix? Can we go ahead along these lines, and construct a matrix P such that Pw=0 for all w∈W?

disgynna0vv · Accepted Answer

Consider a basis of W to be the vectors {w1,…,wk}. Now take the homogeneous system with matrix the row vectors (w1,…,wk). If you take the space of solutions of this system and find a basis for it then you got your desired matrix. We can make this a bit more explicit. The matrix with the row vectors (w1,…,wk) has size k×n. Say it is in block form (A,B) where A is non-singular k×k, and B and k×n−k. Solving the homogeneous system given by the matrix (A,B) expresses the first k components in terms of the the last n-k components ( invert A, ...). To find the basis for the space of solutions, just check that [A,B] ˙[−A−1BIn−k] Therefore, we can take the transpose of the matrix [A,B] ˙[−A−1BIn−k]and find a desired homogenous system.

Yosef Krause · Accepted Answer

I figured something out myself, so I&#039;ll post it. Let {w1,w2,…,wk} be a basis of W and let&#039;s extend this set to a basis of Fn, to obtain {w1,w2,…,wn}юNow, if we define a linear map 𝕟𝕟T:Fn→Fn, such that T(wi)=0 for 1≤i≤k and T(wj)=wj for k+1≤j≤n. As a side-note, we can see that dim(null(T))=k &amp;amp; dim(range(T))=n−k. Consider the matrix A corresponding to this linear map T. Clearly, Ax=0 is the desired system of homogeneous equations!It remains to verify that this construction of A actually works, i.e. the solution space of Ax=0 is W and only W - but I&#039;ll not include that here for brevity.

Task: For n \in \mathbb{N} and W\leq\mathbb{F}^n, show that there exists a homogeneo

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