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Propositions from Kernel, image and rank

All propositions from Kernel, image and rank.

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[PROPOSITION]

Column independence and nullity

The columns of a matrix $A$ are linearly independent if and only if $\operatorname{nullity}(A)=0$ .

A linear dependence among the columns is exactly a non-trivial solution of $Ax=0$ . Thus the columns are independent precisely when the kernel contains only the zero vector.

[1]:Nullity
[2]:Linear independence
[3]:Diagonalisation of a stress matrix

This page builds on Subspaces (Vector spaces), Linear transformations, Basis and dimension (Vector spaces) and Linear independence (Vector spaces).

It leads to Seen Questions, Rank-nullity, Matrix representations, Eigenvalues and eigenvectors (Eigenvalues) and Seen Questions (Eigenvalues).

Part of the thread Kernels and solution sets, in 6 places.

Every solution of $T(x)=b$ is one particular solution plus an element of $\ker T$ .

ALGCAL23

DFKernelthe kernelALG 3 / Here

Propositions from Kernel, image and rank

Column independence and nullity

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