Composition of Linear Transformation and Matrix Multiplication2

Theorem 2.12: Let A be an mxn matrix, B and C be nxp matrices, and D and E b qxm matrices. Then
(d.) If V is an n-dimensional vector space with an ordered basis B, then [IV]B = In.

My question: What does [IV]B mean? Is this the identity matrix with respect to the vector space V which is with respect to the basis B-I'm not sure what that means. Could someone explain this in as much detail possible.

Thanks,

JL
 

HallsofIvy

Science Advisor
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Yes, that is exactly what it means. Specifically, it is saying that if IV(x)= x is the "identity" linear transformation on vector space V, then it is represented by the same matrix no matter what basis you use and that matrix is the n by n matrix with "1"s down then main diagonal and "0"s everywhere else, In.

(Surely, "Let A be an mxn matrix, B and C be nxp matrices, and D and E b qxm matrices" relates to something else. There is no "A", "C", "D" or "E" in what you give and B is NOT a matrix.)
 

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