ehsan_thr said:suppose that {ei} 1<=i<=n is a basis for v and w is a subspace of v of dimension m<n ... . so Can we always find a basis for w which includes m elements of {ei} 1<=i<=n ?
does always w include m elements of {ei} 1<=i<=n ?
No. For example, let v be the set of linear polynomials and let {ei}= {x- 1, x+ 1}. Let w be the set of all multiples of x. Then w is a one dimensional subspace of v and any basis for w must be {ax} for some non-zero number a. None of those is in {ei}.
The other way is true: given any basis for subspace w we can extend it to a basis for v.
T 01:59 PM
The basis of subspace is a set of vectors that span the entire subspace and are linearly independent, meaning none of the vectors can be expressed as a linear combination of the others.
The basis of subspace is important because it allows us to represent any vector within the subspace using a linear combination of the basis vectors. This makes it easier to manipulate and understand the vectors within the subspace.
To find the basis of subspace, you can use the process of elimination to determine which vectors are linearly independent. Another method is to use the Gram-Schmidt process, which involves orthogonalizing a set of vectors to obtain a basis.
Yes, a subspace can have multiple bases. However, all bases for a given subspace will have the same number of vectors, known as the dimension of the subspace.
The number of vectors in the basis of a subspace is equal to the dimension of the subspace. This means that the basis is a minimal set of vectors needed to span the entire subspace.