Re: [線代] linear transformation (onto)
※ 引述《mqazz1 (無法顯示)》之銘言：
: a linear transformation L: V -> W is said to map V onto W if L(V)=W
: show that the linear transformation L defined by
: L(x) = ( x1, x1+x2, x1+x2+x3 )^T
: maps R^3 onto R^3
: 請問這題要怎麼證呢?
: 謝謝!!
L : R^3 → R^3
Let x = [x1,x2,x3]^T , so that
[x1 ] [1 0 0][x1] [1 0 0]
L(x) = [x1+x2 ] = [1 1 0][x2] = Ax ,where A = [1 1 0] for all x in R^3
[x1+x2+x3] [1 1 1][x3] [1 1 1]
Thus L(R^3) = {Ax : x belong R^3} Since A is linear independent,
so L(R^3) = R^3
這樣應該可以吧？
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