Kernels of Linear Maps and Matrices

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Kernel of a linear map

For $L:V\to W$, $\ker(L)=\{v\in V\mid L(v)=0_W\}$: the domain vectors mapped to the codomain's zero vector.

Why is the kernel a subspace?

It contains $0_V$, and linearity makes it closed under addition and scalar multiplication.

When do two vectors have the same image under $L$?

Exactly when their difference lies in the kernel: $L(v_1)=L(v_2)$ if and only if $v_1-v_2\in\ker(L)$.

How does the kernel characterize injectivity?

$L$ is injective if and only if its kernel is trivial: $\ker(L)=\{0_V\}$.

Rank–nullity theorem

For finite-dimensional $V$, $\dim(\operatorname{im}(L))+\dim(\ker(L))=\dim(V)$.

Quiz questions

1. For a matrix $A$ with $n$ columns, which set is its kernel?Answer: The vectors $x\in K^n$ satisfying $Ax=0$.
2. For a real matrix $A$, how is its kernel related to its row space under the standard inner product?Answer: It is the orthogonal complement of the row space.
3. For a real matrix $A$, which description gives its left null space?Answer: The vectors $y$ satisfying $A^Ty=0$.
4. If $Ax=b$ has a particular solution $v$, how can all its solutions be expressed?Answer: As $v+z$, where $z\in\ker(A)$.
5. Why can a numerical computation report a different kernel when a matrix is nearly rank deficient?Answer: Rounding and the tolerance for treating small values as zero can affect the result.