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Chapter 4: Q54E (page 177)

Show that if W is a subspace of an n-dimensional linear space V, then W is finite dimensional as well, and dim(W)鈮�n.

Short Answer

Expert verified

The solution isdim(W)鈮�n.

Step by step solution

01

Explanation of the solution

Consider that V is a vector space of dimension n.

Then there exist a basis B with n vectors to V.

W is a subspace of V.

So, W is also a vector space under the same operations that of V.

Then there exists a basis C to W.

Since, C is the basis of W wit k vectors.

So, C is the linearly independent subset of W.

02

Draw the conslusion

Since, W is the subspace of V, C is the linearly independent subset of V.

So, C either forms a basis for V or it can be extended to form a basis for V.

If C forms the basis to V, then k=n.

Otherwise, k=n.

Putting these things together as follows.

dim(W)鈮�n

Thus, W is finite dimensional.

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