Kernel and Range — Topic Summaries
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Linear Algebra 35 | Rank-Nullity Theorem [dark version]
Rank–nullity theorem is the organizing rule behind how linear maps “trade” dimensions: for any linear map (equivalently, any matrix) from an...
Abstract Linear Algebra 29 | Rank gives Equivalence
Equivalent matrices—those related by changes of bases in the domain and codomain—can represent the same linear map even though their entries differ....
Linear Algebra 38 | Set of Solutions [dark version]
For a linear system written as Ax = B, the solution set is either empty or forms an affine (shifted) subspace: once at least one solution exists,...
Abstract Linear Algebra 31 | Solutions for Linear Equations
Solving a linear equation in an abstract setting boils down to two geometric objects: the range (for whether solutions exist) and the kernel (for...