Easy questions

How is LA 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 LA a transformation? Is it just a definition (notation wise) or is there more to it?


Thanks,

JL
 

HallsofIvy

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How is LA 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 LA 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 "LA". How is it defined?
 
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If A is an m*n matrix, then the mapping [tex]L_A[/tex] is from F^m to F^n and is defined by [tex] L_A(x) = Ax [/tex]. If [tex] x_1 [/tex] and [tex] x_2 ][/tex] are vectors in F^n, then [tex] L_A (x_1 + x_2) = A(x_1 + x_2) = Ax_1 + Ax_2 = L_A(x_1) + L_A(x_2) [/tex]. Also, for any vector x in F^n and any scalar c, we have [tex] L_A(cx) = A(cx) = cAx = cL_A(x) [/tex]. Thus [tex] L_A [/tex] is a linear transformation.
 
If A is an m*n matrix, then the mapping [tex]L_A[/tex] is from F^m to F^n and is defined by [tex] L_A(x) = Ax [/tex]. If [tex] x_1 [/tex] and [tex] x_2 ][/tex] are vectors in F^n, then [tex] L_A (x_1 + x_2) = A(x_1 + x_2) = Ax_1 + Ax_2 = L_A(x_1) + L_A(x_2) [/tex]. Also, for any vector x in F^n and any scalar c, we have [tex] L_A(cx) = A(cx) = cAx = cL_A(x) [/tex]. Thus [tex] L_A [/tex] 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) [tex]A(B + C) = AB + AC and (D + E)A = DA + EA. [/tex]
(b) [tex]a(AB) = (aA)B = A(aB) [/tex] for any scalar a.
(c) [tex]I_mA = A = AI_n[/tex]
(d) If V is an n-dimensional vector space with an ordered basis J, then [tex] [I_V]_J = I_n[/tex]

THanks again.
 

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