Final answer:
To calculate the distance between a subspace and a vector in linear algebra, you can use the projection formula. The projection of a vector onto a subspace is the vector in the subspace that is closest to the given vector. The distance between the vector and the subspace is the length of the difference between the vector and its projection. Here are the steps to calculate the distance: Find the orthogonal projection of the vector onto the subspace. Subtract the projection from the vector to get the difference vector. Calculate the length of the difference vector using the norm or magnitude formula.
Step-by-step explanation:
To calculate the distance between a subspace and a vector in linear algebra, you can use the projection formula. The projection of a vector onto a subspace is the vector in the subspace that is closest to the given vector. The distance between the vector and the subspace is the length of the difference between the vector and its projection. Here are the steps to calculate the distance:
- Find the orthogonal projection of the vector onto the subspace.
- Subtract the projection from the vector to get the difference vector.
- Calculate the length of the difference vector using the norm or magnitude formula.
For example, if the subspace is spanned by the vectors u and v, and the vector you want to find the distance from is w, the projection of w onto the subspace is:
p = ((w · u)/(u · u))u + ((w · v)/(v · v))v
Then, the difference vector is:
d = w - p
Finally, calculate the distance:
distance = ||d||