Linear maps: the "hard core" of linear… algebra

The image and kernel are two particularly important vector spaces—indeed, they are fundamental concepts in linear algebra. For a linear map f that maps each vector $\vec{u}\backslash vec\{u\}$ in the vector space E₁ to the vector $\overrightarrow{v}=f(\overrightarrow{u})\backslash overrightarrow\{v\}\; =\; f\; (\backslash overrightarrow\{u\})$ in the vector space E₂, the kernel is defined as the vector subspace of E₁ consisting of the elements $\vec{u}\backslash vec\{u\}$ such that $f(\overrightarrow{u})=\overrightarrow{0}f\; (\backslash overrightarrow\{u\})\; =\; \backslash overrightarrow\{0\}$. As its name suggests, the image is the vector subspace of E₂ consisting of the vectors $\vec{v}\backslash vec\{v\}$ that are values of the linear map f; that is, there is a vector $\vec{u}\backslash vec\{u\}$ in E₁ such that $\overrightarrow{v}=f(\overrightarrow{u})\backslash overrightarrow\{v\}\; =\; f\; (\backslash overrightarrow\{u\})$.

Natural though these definitions may be, they are terribly abstract!

Yet these concepts are ubiquitous in linear algebra. If f is an endomorphism, its image, denoted Im(f), and its kernel, denoted Ker(f), are vector subspaces (of E₂ and E₁, respectively). For example, in finite-dimensional spaces, Ker(f) consists solely of the zero vector if and only if f is a bijection… But let us not reel off the properties of the image and kernel: even this special issue would not be enough… Instead, let us use a game to see what they mean.