By S. E. Payne

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Cn in F for which v = ni=1 ci vi . The column vector [v]B = (c1 , . . , cn )T ∈ F n is then called the coordinate matrix for v with respect to the basis B, and we may write n v= i=1 ci vi = (v1 , . . , vn ) c1 c2 .. = B[v]B . cn Perhaps we should discuss this last multiplication a bit. In the usual theory of matrix manipulation, if we want to multiply two matrices A and B to get a matrix AB = C, there are integers n, m and p such that A is m × n, B is n × p, and the product C is m × p.

Definition Two n × n matrices A and B over F are said to be similar (written A ∼ B) if and only if there is an invertible n × n matrix P such that B = P −1 AP . You should prove that “similarity” is an equivalence relation on Mn (F ) and then go on to complete the details giving a proof of the following corollary. 3. If A, B ∈ Mn (F ), and if V is an n-dimensional vector space over F , then A and B are similar if and only if there are bases B1 and B2 of V and T ∈ L(V ) such that A = [T ]B1 and B = [T ]B2 .

Then suppose that T (u1 ) = T (u2 ) for u1 , u2 ∈ U . Then by the linearity of T we have T (u1 − u2 ) = T (u1 ) − T (u2 ) = 0, so u1 − u2 ∈ null(T ) = {0}. Hence u1 = u2 , implying T is injective. The following Theorem and its method of proof are extremely useful in many contexts. 2. If U is finite dimensional and T ∈ L(U, V ), then Im(T ) is finite-dimensional and dim(U ) = dim(null(T )) + dim(Im(T )). Proof. (Pay close attention to the details of this proof. ) Start with a basis (u1 , . . , uk ) of null(T ).

### A Second Semester of Linear Algebra by S. E. Payne

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