9×a

9×a denotes the product of the scalar 9 with the element a. In standard algebra, this is written as 9a and means that a is scaled by a factor of 9. The symbol a can represent a real or complex number, a vector, a matrix, or, more generally, an element of a module or vector space where scalar multiplication by 9 is defined. The two notations 9×a and 9a are equivalent in any context where multiplication by a real scalar is defined.

When a is a real or complex number, 9×a equals 9a, the result obtained by multiplying the value of a by 9. This is a linear operation: it preserves addition and scalar multiplication, so 9×(b + c) = 9b + 9c and 9×(k·a) = k×(9a) for any scalar k.

If a is a vector or a matrix, 9×a denotes scalar multiplication, meaning each component or entry of a is multiplied by 9. For a vector v, 9v scales every coordinate by 9. For a matrix A, 9A scales every element of A by 9. This operation commutes with the scalar 9 itself and interacts with addition and matrix multiplication according to the usual rules of scalar multiplication.

In modular arithmetic or other algebraic structures where a is defined, 9×a is interpreted within that structure, e.g., 9a mod n in a ring of integers modulo n. If a is a function, 9×a can be understood as the function that maps each input to 9 times the original function’s value.

Notes: The expression assumes 9×a is defined in the given context; otherwise, its meaning may be undefined.