Prove that if W1 and W2 are subspaces of V such that dim W1

Prove that, if W1 and W2 are subspaces of V such that dim W1 = dim W2, then there exists an orthogonal transformation T such that T(W1) = W2.

Solution

Dim W1 = dim W2 =n

This implies that any vector v in
V can be represented as linear combination of vectors in W1.

i.e.

v = a1w1+a2w2+a3w3+...+anwn

As W2 is a subspace of V,

any vector w\' in W2 can be represented as

w\' = a1\'w1+...+an\'wn

As dim of w\' is n, and bases consist of w1\', w2\'.....

then w\' = already in the linear combination of w1\', w2\'.....

As there are n in both, obviously there must be a linear transformation for w1, w2....wn

such that T(W1) = W2.

If possible let us assume that no T exists.

Then any vector in W2 cannot be represented as linear combination n basis vectors in W1 contradicting the fact that W2 is the subspace and W1 is of dimension n.

Thus proved