3.4 The matrix of linear transformation

# 3.4 The matrix of linear transformation - The matrix of...

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The matrix of linear transformation Theorem 1. Let L : V W be a linear transformation of n ¿ dimentional space V into an m ¿ dimentional space W ( n≠ 0 m ≠ 0 ) and let S ={ v 1 ,v 2 ,…, v n }∧ T ={ w 1 ,w 2 ,…,w m } be bases for V and W , respectively. Then the m×n matrix A , whose j th column is the coordinate vector [ L ( v j ) ] T of L ( v j ) with respect to T , is associated with L and has the following property: If x is in V , then [ L ( x ) ] T = A [ x ] S , where [ x ] S and [ L ( x ) ] T are the coordinate vectors x and L ( x ) with respect to the respective bases S T . Definition. The matrix A of theorem 1 is called the matrix representing L with respect to the bases S T , or the matrix of L with respect to S T . The procedure for computing the matrix of a linear transformation L : V W with respect to the bases S ={ v 1 ,v 2 ,…, v n }∧ T ={ w 1 ,w 2 ,…,w m } for V and W , respectively, is as follows. Step 1. Compute L ( v j ) for j = 1,2, …,n .

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