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Basic properties of linear transformations

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A linear transformation maps 0 to 0.
A linear transformation of a linear combination is the linear combination of the linear transformation
A linear transformation is determined by its action on a basis.
Definition of how the action of a linear transformation on a basis extends to the whole space
The solutions of a homogeneous system are the pre-image (of 0) of a linear transformation.

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The solutions of a homogeneous system are the pre-image (of 0) of a linear transformation.

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The solutions to a nonhomogeneous system are given by a particular solution plus the solutions to the homogeneous system.Close match
The solutions to a homogeneous system of linear equations is a vector space.Close match
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The inverse image of a subspace under a linear transformation is a subspace.Close match
The preimage of a vector is a translation of the kernel of the linear transformationClose match
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A homogeneous system has a nontrivial solution if and only if it has a free variable.Close match
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Definition of solution vector of a linear systemClose match
Vector form of solutions to a consistent linear systemClose match
Definition of pre-image of linear transformationClose match
Definition of pre-image (of a point) under a linear transformationClose match
A linear transformation maps 0 to 0.Close match
The inverse of a linear transformation is a linear transformationClose match
The solutions to a homogeneous linear differential equation is a vector space.Close match
Equivalence theorem for injective linear transformations: T(x)=0 only for x=0.Close match
Equivalence theorem for injective linear transformations: The kernel of T is 0.Close match
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