Question about problem related to orthogonal complements

[starcoast]

Homework Statement

If V is the orthogonal complement of W in Rⁿ, is there such a matrix with row space V and nullspace W? Starting with a basis for V, construct such a matrix.

The Attempt at a Solution

I've been trying to use the fact that V is the left nullspace of the column space of W to get to a matrix with row space V but I haven't been able to get anywhere. A push in the right direction would be really helpful right now.

 

[mighty2000]

Thank you for your question. I can provide some insights on how to construct such a matrix. First, let's define some terms to make sure we are on the same page. The orthogonal complement of W in Rn is the set of all vectors in Rn that are perpendicular to every vector in W. In other words, the dot product of any vector in V and any vector in W is equal to zero.

Now, to construct a matrix with row space V and nullspace W, we can start by choosing a basis for V. Let's call this basis {v1, v2, ..., vk}. Since V is the orthogonal complement of W, we know that any vector in V is perpendicular to every vector in W. This means that the dot product of any vector in V with any vector in W is equal to zero. We can use this fact to construct the rows of our matrix.

For the first row, we can choose any vector in W and set its dot product with v1 to be equal to zero. Similarly, for the second row, we can choose another vector in W and set its dot product with v2 to be equal to zero. We can continue this process for all k rows, using different vectors in W each time. This will result in a matrix with row space V and nullspace W.

I hope this helps. Let me know if you have any further questions or if you need clarification on any of the steps. Good luck with your problem!