3 Linear Maps

3.B Null Spaces and Ranges

NullSpace And Injectivity

We begin with the set of vectors that get mapped to 0.

3.12 Definition nullsapce, null $T$

For $T\in \mathcal{L}(V, W)$, the null space of T,

$$\text{null}\ T = \{v\in V : Tv = 0\}$$

3.14 Theorem The null space is a subspace

For $T\in \mathcal{L}(V, W)$. Then $\text{null}\ T$ is a subspace of $V$.

3.15 Definition Injective

A function $T: V \rightarrow W$ is call injective if $Tu=Tv$ implies $u=v$.

• $T$ is injective if $u\not ={v}$ implies $Tu\not =Tv$
• $T$ is injective if it maps distinct inputs to distinct outputs
• one-to-one equals injective

3.16 Theorem Injectivity is quivalent to nullspace equals $\{0\}$

For $T\in \mathcal{L}(V, W)$. Then $T$ is injective if and only if $\text{null}\ T = \{0\}$.

Thus we can check whether a linear map is injective by checking whether 0 is the only vector that mapped to 0 by 3.16.

Range and Surjectivity

We give a name to the set of output vectors of a function.

3.17 Definition range

For $T\in \mathcal{L}(V, W)$, the range of $T$,

$$\text{range}\ T = {Tv:v\in V}$$

• image euqals to range.

3.19 Theorem The range is a subspace

For $T\in \mathcal{L}(V, W)$. Then $\text{range}\ T$ is a subspace of $W$.

3.15 Definition Surjective

A function $T: V \rightarrow W$ is call surjective if $\text{range}\ T = W$

• onto equals injective

Fundamental Theorem of Linear Maps

3.22 Theorem Fundamental Theorem of Linear Maps

Suppose $V$ is finite-dimensional and $T\in \mathcal{L}(V, W)$. Then $\text{range}\ T$ is finite-dimensional and

$$\text{dim}\ V = \text{dim}\ \text{null}\ T + \text{dim}\ \text{range}\ T$$

3.23 A map to a smaller dimensional space is not injective

Suppose $V$ and $W$ are finite-dimensional vector spaces such that $\text{dim}\ V > \text{dim}\ W$. Then no linear map from $V$ to $W$ is injective.

3.24 A map to a larger dimensional space is not surjective

Suppose $V$ and $W$ are finite-dimensional vector spaces such that $\text{dim}\ V < \text{dim}\ W$. Then no linear map from $V$ to $W$ is surjective.

Examples

homogeneous system of linear equations

Let $A_{j,k} \in \bold{F}$ for $j = 1,...,m$ and $k=1,...,n$. Consider the homogeneous system of linear equations

$$\sum_{k=1}^nA_{1,k}x_k = 0 \\ \vdots \\ \sum_{k=1}^nA_{m,k}x_k = 0$$

$x_1=···=x_n$ is a solution. Question: Whether any other solutions exist?

Define $T:\bold{F}^n \rightarrow \bold{F}^m$ by

$$T(x_1, ..., x_n) = (\sum_{k=1}^nA_{1,k}x_k, ..., \sum_{k=1}^nA_{m,k}x_k)$$

When $\text{null}\ T > 0$, we have nonezero solutions. By 3.16, we have $T$ is not injective.

• Rephrase question: What condition ensures that $T$ is not injective?
• Answer: By 3.22, if $n > m$, then $T$ is not injective.

Inhomogeneous system of linear equations

Let $A_{j,k} \in \bold{F}$ for $j = 1,...,m$ and $k=1,...,n$. For $c_1,...,c_m \in \bold{F}$,

$$\sum_{k=1}^nA_{1,k}x_k = c_1 \\ \vdots \\ \sum_{k=1}^nA_{m,k}x_k = c_m$$

Question: Whether this is some choice of $c_1,..., c_m \in \bold{F}$ such that no solution exists.

Define $T:\bold{F}^n \rightarrow \bold{F}^m$ by

$$T(x_1, ..., x_n) = (\sum_{k=1}^nA_{1,k}x_k, ..., \sum_{k=1}^nA_{m,k}x_k)$$

We want $\text{range}\ T\not= \bold{F}^m$, it means $T$ is not surjective.

• Rephrase question: What condition ensures that $T$ is not surjective?
• Answer: By 3.22, if $n < m$, then $T$ is not surjective.