What does a linear combination of vectors imply about their associated images?

A linear combination of vectors refers to a mathematical operation where each vector is multiplied by a corresponding scalar and then summed together. When considering the associated images of vectors under a linear transformation, the impact of these coefficients is significant.

Specifically, if you have vectors ( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n ) and you take a linear combination, say ( c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \ldots + c_n \mathbf{v}_n ), it translates to their images under a linear transformation ( T ) being combined in the same manner. That is, if ( T(\mathbf{v}_i) ) is the image of ( \mathbf{v}_i ), then the image of the linear combination can be expressed as ( T(c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \ldots + c_n \mathbf{v}_n) = c_1 T(\mathbf{v}_1) + c_2 T(\mathbf{v}_2) + \ldots + c_n T(\mathbf{