
Subspace
Especially, let , we obtain the basis of . The basis is , where , . So all the points in can be expressed as for any . Thus represents a subspace.

isomorphism
They constitute the basis of the given subspace of .
v) It is a subspace, since it is closed under the addition and the product by a scalar. To find its basis, let us consider any polynomial from . Then .

Matrix subspace
So is a subspace of generated by the vector and its basis is . This provides an example of working with a matrix to complete a proof regarding subspaces.

Hahn Banach Theorem Application
Prove that each finite dimensional subspace of is complemented in .
Hint: Suppose that M is a finite dimensional subspace of and is a basis for M. Define linear functionals : M K by for in K, k=1,2,...,n.

Vector Subspace, Orthonormal Basis, Othogonal Projection and Inner Product
158492 Vector Subspace, Orthonormal Basis, Othogonal Projection and Inner Product Let W be the subspace of R^2 spanned by the vector (3, 4). Using the standard inner product, let E be the orthogonal projection of R^2 onto W.

Let a ε complex numbers and distinct from 0.
Prove that the eigenvectors of the transformation T : Complex numbers^2 > complex numbers^2 given by T ( z , w ) = ( z + aw , w) span on 1dimensional subspace of complex numbers^2 and give a basis for this subspace.

Vector Subspaces, GramSchmidt, Orthogonal, Basis
Find the distance from y to the subspace of R4 spanned by v1 and v2.
#3) The given set is a basis for a subspace W. Use the GramSchmidt process to produce an orthogonal basis for W.

Linear Algebra : Span and Subspace
Since is a basis of V, every vector of (in V) can be expressed as a linear combination of . This is impossible, as is a basis of V and so are linear independent. So, .
Question 9.
Proof.

Orthogonal and orthonormal basis
The GramSchmidt Theorem:
Given a basis for subspace W of Vn define:
Then is an orthogonal basis of W and has the same span as
We shall use the standard inner product:
In our case:
And:
In the next step

Vector Spaces, Basis and Quotient Spaces
97544 Vector Spaces, Basis and Quotient Spaces See the attached file.
1. Let and be vector spaces over and let be a subspace of
Show that for all is a subspace of
and this subspace is isomorphic to .