MHB Linear Algebra: Kernel & Range of Linear Transformation

matqkks · Oct 3, 2012

Why are we interested in looking at the kernel and range (image) of a linear transformation on a linear algebra course?

Siron · Oct 4, 2012

They can be useful. Suppose you have a linear map f: V \to W. If you want to know if this linear map is injective (i.e one-to-one map) then you can take a look at the kernel: $$\ker( f)=\{0\} \Leftrightarrow \ f \ \mbox{is injective}$$

There's also the following result.
$$V/\mbox{ker}( f) \cong \mbox{Im}( f)$$
which can be very useful because it's easier to work with the image in stead of the quotientspace $V/\mbox{ker}( f)$

These are offcourse a lot of other results but these two are the first I could remember immediately.

MHB Linear Algebra: Kernel & Range of Linear Transformation

Thread 'When are Markov Matrices also Martingales?'

Thread 'The colorful world of ##2\times 2## complex matrices'

Similar threads

Hot Threads

I How to show ##p(x)=g(x)x\pm 1\in\Bbb{Q}[x]## is irreducible in ##\Bbb{Q}_{\Bbb{Z}}[x]##?

I Showing ##k[x_1,\ldots,x_n]/\mathfrak{a}## is finite dimensional

I How do we distinguish two different notations for cokernel and coimage?

I Localising a non integral domain at a prime

I ##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?

Recent Insights

Insights Thinking Outside The Box Versus Knowing What’s In The Box

Insights Why Entangled Photon-Polarization Qubits Violate Bell’s Inequality

Insights Quantum Entanglement is a Kinematic Fact, not a Dynamical Effect

Insights What Exactly is Dirac’s Delta Function? - Insight

Insights Relativator (Circular Slide-Rule): Simulated with Desmos - Insight

Insights Fixing Things Which Can Go Wrong With Complex Numbers