Re: [線代] 幾題考題請教
※ 引述《killeress (彌朧澄昏)》之銘言：
: 1.Please find the dimension and a basis for the each of four fundamental
: subspace (row space column space null space and left null space) of matrix
: | 1 2 1 3 2 |
: | 3 4 9 0 7 |
: | 2 3 5 1 8 |
: | 2 3 5 1 8 |
: | 2 2 8 -3 5 |
: 主要問題:自己看書沒看到類似習題，大約只知道dimension為basis的數目
: 找的到basis應該就可以找到dim? left null space的定義也找不到
: 不知道從何下手
對
left null space 左乘矩陣 = 0 (不負責亂猜)
: 2.Let T:R^4 ->R^3 be a linear transformation defined by
: T(x,y,x,t)= (x-2y+z-t ,3x-2z+3t,5x-4y+t )
: 1.Find bases for kernel and image T
: 2.Find the value of w if (1,5,w) 屬於 ImT ,the image of T.
β= {(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)}
γ= {(1,0,0),(0,1,0),(0,0,1)}
γ [1 -2 1 -1] [1 -2 1 -1]
=> [T] = [3 0 -2 3] -> [0 6 -5 6]
β [5 -4 0 1] [0 0 0 0]
xεker(T)
x_1-2x_2+ x_3 -x_4=0
6x_2-5x_3+6x_4=0
[2/3t-s]
[5/6t-s]
x=[ t ] Thus, ker(T)=span{(2/3,5/6,1,0),(-1,-1,0,1)}#
[ s]
R(T) = span{(1,3,5),(-2,0,-4),(1,2,0),(-1,3,1)}
= span{(1,0,2),(0,1,1)}
(1,5,w) = 1(1,0,2)+5(0,1,1) => w = 7#
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