Final answer:
The correct statement is a. A(u+v) = Au + Av. When a matrix is multiplied by the sum of two vectors, the result is equal to the sum of the matrix multiplied by each vector separately.
Step-by-step explanation:
The correct statement is a. A(u+v) = Au + Av. When a matrix is multiplied by the sum of two vectors, the result is equal to the sum of the matrix multiplied by each vector separately. This can be demonstrated using an example:
Let's say we have a 2x2 matrix A, given by:
A = [ a11 a12 ]
[ a21 a22 ]
And two vectors u and v in R^n:
u = [u1]
[u2]
v = [v1]
[v2]
The result of multiplying the matrix by the sum of the vectors is:
A(u+v) = [ a11*u1 + a12*v1 ]
[ a21*u2 + a22*v2 ]
On the other hand, if we multiply the matrix by each vector separately and then sum up the results, we get:
Au + Av = [ a11*u1 + a12*u1 ] + [ a11*v1 + a12*v2 ]
[ a21*u2 + a22*u2 ] [ a21*v1 + a22*v2 ]
As we can see, both results are the same, which confirms that the statement a is true. The other statements b, c, and d are not true and can be disproven using similar examples.