The essence of linear algebra 4 matrix multiplication and linear compound transformation

Catalog

4.1 The essence of matrix

4.2 Composite transformation

 4.3 Multiplication of matrices

1. Matrix multiplication formula  

2. The order of matrix multiplication

3. The associative law of matrix multiplication

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4.1 The essence of matrix

• The first column represents the transformed i Cap vector
• The second column represents the transformed j Cap vector
• The whole matrix represents the rule of linear transformation of a vector

The multiplication of a matrix is to let a vector [x,y] After the linear transformation rules defined by the matrix, the objective vector is obtained

  

4.2 Composite transformation

I want to make a compound transformation of a vector ： Rotate then cut , There are two ways ：

1.   First, the original vector is left multiplied by the rotation matrix to obtain the matrix after rotation transformation , Then left multiply the cut matrix to get the cut matrix value , Matrix multiplication is divided into two steps to perform the operation of composite transformation
2. After the compound transformation i and j vector , Put them together into a composite matrix , Let the original vector be multiplied by it

 

Further, it can be concluded that ： Multiply two transformation matrices = Composite matrix

Multiply two matrices ： Two linear transformations act on the vector successively

Be careful This multiplication needs to be read from right to left , Can be equivalent to Compound function Writing , Write down the operation first , After the operation, write it on the outside  

 4.3 Multiplication of matrices

1. Matrix multiplication formula  

•   First of all to see M1 The first column of the matrix , Equivalent to the transformed i vector , Let it relate to M2 Multiply , Get the first column of the composite matrix
• Look again M1 The second column of the matrix , Equivalent to the transformed j vector , Let it relate to M2 Multiply , Get the second column of the composite matrix
• Then we can get the calculation formula of the matrix , It is essentially a linear transformation of a vector

2. The order of matrix multiplication

Be careful ： The order in which the matrices are multiplied affects the final result , Because the geometric meaning of matrix multiplication is the transformation of vectors  , So the order of transformation is different , The corresponding transformation results must be different

3. The associative law of matrix multiplication

1. Matrix multiplication satisfies the law of union , That is, the position of brackets does not affect the result of matrix multiplication , Because wherever you put parentheses , All are 3 Use the same transformation 3 The effect of this action in the same order

2. First apply C Transformation , Then apply B Transform sum A Transformation == First apply C Transform sum B Transformation , Then apply A Transformation