Minal said:An orthogonal basis set spanning R4 has four vectors, v1, v2, v3 and v4.
If v1 and v2 are
[ −1 2 3 0 ] and [−1 1 −1 0 ]
find v3 and v4.
Please explain this in a very simple way.
Minal said:please write the formula to find v3 and v4.
To determine if a set of vectors is an orthogonal basis for R4, you must check if all the vectors are orthogonal (perpendicular) to each other and if they span the entire space of R4. This means that each vector in the set must be orthogonal to every other vector and the linear combination of the vectors must be able to represent any vector in R4.
The purpose of finding vectors in an orthogonal basis set is to make calculations and operations on vectors in R4 easier. Since orthogonal vectors are perpendicular, they have zero dot product with each other, making it simpler to calculate projections, find angles, and solve systems of equations.
To find the orthogonal basis set for R4, you can use the Gram-Schmidt process. This involves taking a set of linearly independent vectors in R4 and using orthogonalization to create a new set of vectors that are orthogonal to each other. The resulting set will be an orthogonal basis set for R4.
Yes, there can be multiple orthogonal basis sets for R4. This is because there are different ways to choose a set of linearly independent vectors and apply the Gram-Schmidt process to create an orthogonal basis set. However, all of these sets will have the same number of vectors and will span the same space of R4.
Orthogonal vectors are always linearly independent, but the reverse is not always true. If a set of vectors is linearly independent, then they will be orthogonal as well. However, there can be sets of vectors that are orthogonal but not linearly independent. This is because orthogonal vectors can have zero dot product with each other, but still be able to create a linear combination to represent any vector in R4.