Understanding LA: Linear Transformation of Matrix A

[jeff1evesque]

How is L_A a linear function? What kind of operation is action on A? I thought L denotes a linear transformation. So if we have a matrix A, how is the L_A a transformation? Is it just a definition (notation wise) or is there more to it?


Thanks,

JL

 

[HallsofIvy]

How is L_A a linear function? What kind of operation is action on A? I thought L denotes a linear transformation. So if we have a matrix A, how is the L_A a transformation? Is it just a definition (notation wise) or is there more to it?


Thanks,

JL

I have no idea what you mean by "L_A". How is it defined?

 

[JG89]

If A is an m*n matrix, then the mapping $$L_A$$ is from F^m to F^n and is defined by $$L_A(x) = Ax$$. If $$x_1$$ and $$x_2 ]$$ are vectors in F^n, then $$L_A (x_1 + x_2) = A(x_1 + x_2) = Ax_1 + Ax_2 = L_A(x_1) + L_A(x_2)$$. Also, for any vector x in F^n and any scalar c, we have $$L_A(cx) = A(cx) = cAx = cL_A(x)$$. Thus $$L_A$$ is a linear transformation.

 

[jeff1evesque]

If A is an m*n matrix, then the mapping $$L_A$$ is from F^m to F^n and is defined by $$L_A(x) = Ax$$. If $$x_1$$ and $$x_2 ]$$ are vectors in F^n, then $$L_A (x_1 + x_2) = A(x_1 + x_2) = Ax_1 + Ax_2 = L_A(x_1) + L_A(x_2)$$. Also, for any vector x in F^n and any scalar c, we have $$L_A(cx) = A(cx) = cAx = cL_A(x)$$. Thus $$L_A$$ is a linear transformation.

THanks, that's exactly what I thought. However, in the text I am reading, it says L_A is linear immediately from theorem 2.12:

Theorem 2.12:
Let A be an mxn matrix, B and C be nxp matrices, and D and E be qxm matrices. Then,
(a) $$A(B + C) = AB + AC and (D + E)A = DA + EA.$$
(b) $$a(AB) = (aA)B = A(aB)$$ for any scalar a.
(c) $$I_mA = A = AI_n$$
(d) If V is an n-dimensional vector space with an ordered basis J, then $$[I_V]_J = I_n$$

THanks again.