Page 60 - Linear Algebra Done Right

Chapter 3. Linear Maps
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Now we can show that no linear map from a finite-dimensional vec-
tor space to a “smaller” vector space can be injective, where “smaller”
is measured by dimension.
3.5 Corollary: If V and W are finite-dimensional vector spaces such
that dim V> dim W, then no linear map from V to W is injective.
Proof: Suppose V and W are finite-dimensional vector spaces such
that dim V> dim W. Let T ∈L(V, W). Then

dim null T = dim V − dim range T
≥ dim V − dim W
> 0,

where the equality above comes from 3.4. We have just shown that
dim null T> 0. This means that null T must contain vectors other
than 0. Thus T is not injective (by 3.2).

The next corollary, which is in some sense dual to the previous corol-
lary, shows that no linear map from a finite-dimensional vector space
to a “bigger” vector space can be surjective, where “bigger” is measured
by dimension.

3.6 Corollary: If V and W are finite-dimensional vector spaces such
that dim V< dim W, then no linear map from V to W is surjective.

Proof: Suppose V and W are finite-dimensional vector spaces such
that dim V< dim W. Let T ∈L(V, W). Then
dim range T = dim V − dim null T
≤ dim V
< dim W,

where the equality above comes from 3.4. We have just shown that
dim range T< dim W. This means that range T cannot equal W. Thus
T is not surjective.

The last two corollaries have important consequences in the theory
of linear equations. To see this, fix positive integers m and n, and let
n
a j,k ∈ F for j = 1,...,m and k = 1,...,n. Define T : F → F m by