Write the subspace spanned by vectors as a kernel of a matrix.

[bobby2k]

Hi

Lets say I have a vectorspace in Rn, that is called V.
V = span{v1,v2,... vk}

Is it then possible to create an m*n matrix A, whose kernel is V.
That is Ax = 0, x is a sollution if and only if x is an element of V.

Also if this is possible, I imagine that k may not b equal to m?

 

[bobby2k]

Also, if it makes the question easier, we can assume that v1,...,vk is linerly independent.

 

[Mark44]

Hi

Lets say I have a vectorspace in Rn, that is called V.
V = span{v1,v2,... vk}

Is it then possible to create an m*n matrix A, whose kernel is V.
That is Ax = 0, x is a sollution if and only if x is an element of V.

Also if this is possible, I imagine that k may not b equal to m?

Also, if it makes the question easier, we can assume that v1,...,vk is linerly independent.

Homework problem?

 

[bobby2k]

Homework problem?

Not really. In my book they talk about a column space for R3. It is obvious that the column space is spanned by two particular vectors([3,3,4] and [-1,-1,5]). But then they make the statement that the columnspace is those vectors who have the property that x1-x3 = 0, this is the same as saying the column space is the vectors who satisfy Av where A = [1 0 -1], so I am wondering how we can do this generally. That is, change the description of a vector space, from a spanning-set, to a Kernel. And if there is an algorithm for doing this.

Another way of asking is to compare with solving the equaion Bx =0. Then we can find vectors that span Null B. I want to go the opposite way, we have the vectors that span Null B, but I want to find the matrix B(I suspect this matrix may not be unique.).

 

[Office_Shredder]

The first thing I think that should be pointed out is that you aren't changing the description of a vector space in general, just giving a different way of describing vector subspaces of Rⁿ. Obviously you can't define a vector space in general as the kernel of a linear transformation because a linear transformation doesn't make sense if you don't know what a vector space is already.

Anyway ,back to the task at hand, if my matrix B has rows b₁,...,b_m, then Bv = 0 means that v is perpendicular to each row of B. So a matrix for which B v_j = 0 for each j has rows which are perpendicular to every v_j, and then to make sure that the kernel does not have a larger subspace than you want you need to make sure B has enough rows in it - I'll let you think a bit on what procedure you can do to generate B algorithmically (it only requires a standard linear algebra result).

 

[bobby2k]

The first thing I think that should be pointed out is that you aren't changing the description of a vector space in general, just giving a different way of describing vector subspaces of Rⁿ. Obviously you can't define a vector space in general as the kernel of a linear transformation because a linear transformation doesn't make sense if you don't know what a vector space is already.

Anyway ,back to the task at hand, if my matrix B has rows b₁,...,b_m, then Bv = 0 means that v is perpendicular to each row of B. So a matrix for which B v_j = 0 for each j has rows which are perpendicular to every v_j, and then to make sure that the kernel does not have a larger subspace than you want you need to make sure B has enough rows in it - I'll let you think a bit on what procedure you can do to generate B algorithmically (it only requires a standard linear algebra result).

Is this proof correct?, I have found a matrix, whose Kernel contains the spanning set, but I do not yet see how to prove that it does not contain any other vectors.

I assume that the vectors v1, vk are linerly independent.

If b is a row vector in B, then we must have

b*v1=0
b*v2=0
.
.
b*vk=0, where * is matrix multiplication(not dot-product)We can transpose this to.

(v1)'*b'=0
.
.
.
(vk)'*b'=0

And we can put it all together in a Matrix.

|v1...|
|v2...|
|...| * b' = 0(vector)
|...|
|vk...|

When we solve this we get n-k linerly independent b', because the rank of the matrix is k, and the rank-theorem gives dim Null(Matrix)=n-k.

So we can put these b row vectors together to form B, which is then (n-k)*n. And we have B*x =0(vector), if x is in span{v1,...,vk}.
But we also have to show that if vector is not in span{v1,..,vk} than Bx is not 0(vector).
To show this i can use contrapositive proof. That is if Bx=0(vector), then x is in span{v1,..,vk}.
This last part is where I get stuck, do you see how to proceed?

 

[Office_Shredder]

For this part I recommend considering the span of $\left{ v_1,...,v_k, b_1,...,b_{n-k} \right}$.

 

[Bipolarity]

First extend the vectors $\{ v_{1} v_{2} ... v_{k} \}$ to $\{ v_{1} v_{2} ... v_{k} v_{k+1} ... v_{n} \}$, a basis for $ℝ^{n}$. Then define the following linear map $T : ℝ^{n} → ℝ^{n}$ by the following operations on that basis:
$T(v_{1}) = 0$
$T(v_{2}) = 0$
...
$T(v_{k}) = 0$
$T(v_{k+1}) = v_{k+1}$
...
$T(v_{n}) = v_{n}$.

Then $[T]_{β}$ where $β$ is the standard ordered basis for $ℝ^{n}$ will have a kernel consisting of V. Try proving this result yourself.

BiP