To prove that (A B) C=A(B C), use the column vectors b1, …. bn of B. First suppose that C has on

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To prove that (A B) C=A(B C), use the column vectors b1, …....

To prove that $(A B) C=A(B C)$, use the column vectors $b_{1}, \ldots . b_{n}$ of $B$. First suppose that $C$ has only one column $c$ with entries $c_{1}, \ldots, c_{n}$ : $A B$ has columns $A b_{1}, \ldots, A b_{n}$ and $B c$ has one column $c_{1} b_{1}+\cdots+c_{n} b_{n}$. Then $(A B) c=c_{1} A b_{1}+\cdots+c_{n} A b_{n}$ equals $A\left(c_{1} b_{1}+\cdots+c_{n} b_{n}\right)=A(B c)$. Linearity gives equality of those two sums, and $(A B) c=A(B c)$. The same is true for all other _ of $C$. Therefore $(A B) C=A(B C)$.

To prove that $(A B) C=A(B C)$, use the column vectors $b_{1}, \ldots . b_{n}$ of $B$. First suppose that $C$ has only one column $c$ with entries $c_{1}, \ldots, c_{n}$ :
$A B$ has columns $A b_{1}, \ldots, A b_{n}$ and $B c$ has one column $c_{1} b_{1}+\cdots+c_{n} b_{n}$.
Then $(A B) c=c_{1} A b_{1}+\cdots+c_{n} A b_{n}$ equals $A\left(c_{1} b_{1}+\cdots+c_{n} b_{n}\right)=A(B c)$.
Linearity gives equality of those two sums, and $(A B) c=A(B c)$. The same is true for all other _ of $C$. Therefore $(A B) C=A(B C)$.

Key Concepts

Linearity

Linearity is a fundamental property of matrix operations, indicating that a matrix acting on a linear combination of vectors yields the same result as the corresponding linear combination of the matrix acting on each vector individually. This property is pivotal in the proof of the associativity of matrix multiplication, as it allows the distribution of the matrix over a sum of vectors, ensuring that the order in which matrices are multiplied does not change the final result.

Column Vector Decomposition

The concept of column vector decomposition in matrix multiplication involves breaking down a matrix into its individual column vectors. This enables one to interpret the multiplication of a matrix A by another matrix B as applying A to each column vector of B separately. Such an approach aids in visualizing and proving properties like associativity by focusing on the transformation of each column.

Associative Property of Matrix Multiplication

The associative property of matrix multiplication states that for any three matrices A, B, and C (with dimensions that allow the products to be defined), the equality (AB)C = A(BC) holds. This property is essential in many areas of mathematics as it allows the grouping of matrices in a product to be rearranged without affecting the outcome, thereby simplifying analysis and proofs in linear algebra.

Matrix Multiplication

Matrix multiplication is a process where each element of the product matrix is obtained by taking the dot product of a row from the first matrix with a column from the second matrix. This operation can be understood through its impact on the column vectors of the second matrix, with each column being transformed by the first matrix to form the corresponding column in the product matrix.

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