a = b - c
b = -2b + d
c = 4d
d = c - 2d
e = 2b + 6dA subspace is a subset of a vector space that satisfies the properties of a vector space. In other words, it is a subset of a larger space that is closed under vector addition and scalar multiplication.
The basis of a subspace is a set of linearly independent vectors that span the subspace. In other words, the basis is the minimum number of vectors needed to represent all other vectors in the subspace through linear combinations.
Finding a basis for a subspace is important because it allows us to understand the structure of the subspace and to perform calculations and transformations on the subspace. It also allows us to find a minimal set of vectors needed to represent the subspace, which can be useful in various applications.
To find the basis for a subspace, we need to find a set of linearly independent vectors that span the subspace. This can be done through various methods, such as Gaussian elimination, finding the null space of a matrix, or using the Gram-Schmidt process.
The dimension of a subspace is equal to the number of vectors in its basis. This means that the basis of a subspace will always have the same number of vectors as the dimension of the subspace. Additionally, any set of linearly independent vectors that span the subspace can be considered a basis for that subspace.