2)a1=(1.0,1,0) B1=(0,1,0.1) a2=(1,1,0,1); B2=(0,1,1,0) a1=(1,0,2,0,) (3){a2=(2,0,1,1) ∫=(331-2 B2=(1,3,0,-3) 1 解:把由向量a生成的子空间项由向量生成的子空间分封记为W1,W2 (1)dim(Wi+W2)=3, dim WinW2=1, W1+W2的基:a1,a2,61 W1∩W2的基(3,-2,3,8)(=3(-2a1+1la2)=-461+3 (2)dim(Wi+W2)=4, dim WinW2=0 W1+W2的基:a1,a2,1,B2; (3)dim(Wi+W2)=3, dim Win W1+W2的基:a1,a2,B1 W1∩W2的基:(2,0,1,1)(=a2=1-B2) 3.设W,W1,W2都闭向量空间V的子空间,且 WICw2, wnwi=wnw2, W+Wi=w+W2 证明:W1=W2 证明:dimW+dimW1=dim(W+W1)+dim(W∩W1) dim w+dim W2= dim(W+W2)+dim(W nw2). 所以上式列端相等.可得dimW1=dimW2.又因W1sW2,所以W1=W2 4.设V1,V闭n维线性空间V的两个子空间,并且满足 dim(Vi+V2)=dim(Vi V2)+1 证明:Ⅵ≤V或VV1 证明:因为dm(V∩v2)≤dimV1≤dim(V1+V2)=dim(V1∩V)+1,两个等号中必有一个成 立.何向下总等号成立,则因V∩vsV,可得V∩V=V,从而VV.何向列总等号成立,则因 V1sⅵ+V,可得Ⅵ=ⅵ1+V,从而vsV 5.设V=K4,a1=(1,2,1,2),a2=(2,1,2,1),W=L(a1,a2).限子空间W在V中的一个补空 间 解:设a3=(0,0,1,0),a4=(0,0.0,1),则因a1,a2,a3,a4线性无关,所以L(0,0.,1,0),(0,0,0,1) 闭W在V中的一个补空间 6.证明:每一个n维线性空间都闭n个一维子空间的直项 证明:设V为n维线性空间,a1,…,an闭V的基.令W=L(a1),则V=W1+W n=dimV=∑=1dimW,所以 V=W1⊕W2⊕…⊕W 7.证明:n维线性空间V的每一个真子空间都闭若干个n-1维子空间的交 证明:设W闭V的真子空间,则r=dimW<dimV=n.取W的一个基a1,……,ar,将其扩充成 V的基a1,……,an,取何下的n=r个n-1维线性子空间 Vi=L(ar(2) ( α1 = (1, 0, 1, 0) α2 = (1, 1, 0, 1); ( β1 = (0, 1, 0.1) β2 = (0, 1, 1, 0); (3)    α1 = (1, 0, 2, 0,) α2 = (2, 0, 1, 1) α3 = (1, 0, −1, 1); ( β1 = (3, 3, 1, −2) β2 = (1, 3, 0, −3). : NN αi *￾pq:N βi *￾pq" W1, W2. (1) dim(W1 + W2) = 3, dim W1 ∩ W2 = 1, W1 + W2 z: α1, α2, β1, W1 ∩ W2 z: (3, −2, 3, 8) ³ = 1 3 (−2α1 + 11α2) = −4β1 + 3β2 ´ ; (2) dim(W1 + W2) = 4, dim W1 ∩ W2 = 0, W1 + W2 z: α1, α2, β1, β2; (3) dim(W1 + W2) = 3, dim W1 ∩ W2 = 1, W1 + W2 z: α1, α2, β1, W1 ∩ W2 z: (2, 0, 1, 1)(= α2 = β1 − β2). 3.  W, W1, W2 m pq V ￾pq, ? W1 ⊆ W2, W ∩ W1 = W ∩ W2, W + W1 = W + W2. ST: W1 = W2. : dim W + dim W1 = dim(W + W1) + dim(W ∩ W1), dim W + dim W2 = dim(W + W2) + dim(W ∩ W2), #$y)aeV. >P dim W1 = dim W2. Q! W1 ⊆ W2, #$ W1 = W2. 4.  V1, V2  n Ft&pq V 7f￾pq, W?-. dim(V1 + V2) = dim(V1 ∩ V2) + 1, ST: V1 ⊆ V2 D V2 ⊆ V1. : !" dim(V1 ∩ V2) 6 dim V1 6 dim(V1 + V2) = dim(V1 ∩ V2) + 1, 7fVY@GHf* +. 8VY*+, J! V1 ∩ V2 ⊆ V1, >P V1 ∩ V2 = V1, C% V1 ⊆ V2. 8VY*+, J! V1 ⊆ V1 + V2, >P V1 = V1 + V2, C% V2 ⊆ V1. 5.  V = K4 , α1 = (1, 2, 1, 2), α2 = (2, 1, 2, 1), W = L(α1, α2). s￾pq W k V HfOp q. :  α3 = (0, 0, 1, 0), α4 = (0, 0, 0, 1), J! α1, α2, α3, α4 t&,*, #$ L((0, 0, 1, 0),(0, 0, 0, 1))  W k V HfOpq. 6. ST: sHf n Ft&pqm n fHF￾pq.:. :  V " n Ft&pq, α1, · · · , αn  V z. I Wi = L(αi), J V = W1 + W2 + · · · + Wn. Q, n = dim V = Pn i=1 dim Wi , #$ V = W1 ⊕ W2 ⊕ · · · ⊕ Wn. 7. ST: n Ft&pq V sHf￾pqmPf n − 1 F￾pq. :  W  V ￾pq, J r = dim W < dim V = n. z W Hfz α1, · · · , αr, v<0* V z α1, · · · , αn. z n − r f n − 1 Ft&￾pq Vj = L(α1, · · · , αj−1, αj+1, · · · , αn), j = r + 1, · · · , n. · 4 ·