设(3(alpha_{1}-alpha)+2 (alpha_{2}+alpha)=5(alpha_{3}+alpha),) 试求向量` alpha=`_____，其中`alpha_{1}=(2,5,1,3)^{T}, alpha_{2}=(10,1,5,10)^{T}`，( alpha_{3}=(4,1,-1,1)^{T}。)

设以下三个向量组\(S_{1}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3} \},\)\(S_{2}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\},\)\(S_{3}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{5}\} \)的秩依次为\(3,3,4,\)则\(S={\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{5}-\alpha_{4} }\)的秩为______ 。

设以下三个向量组\(S_{1}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3} \},\)\(S_{2}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\},\)\(S_{3}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{5}\} \)的秩依次为\(3,3,4,\)则\(S={\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{5}-\alpha_{4} }\)的秩为______ 。

设以下三个向量组\(S_{1}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3} \},\)\(S_{2}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\},\)\(S_{3}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{5}\} \)的秩依次为\(3,3,4,\)则\(S={\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{5}-\alpha_{4} }\)的秩为______ 。

设以下三个向量组\(S_{1}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3} \},\)\(S_{2}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\},\)\(S_{3}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{5}\} \)的秩依次为\(3,3,4,\)则\(S={\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{5}-\alpha_{4} }\)的秩为______ 。

设以下三个向量组\(S_{1}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3} \},\)\(S_{2}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4}\},\)\(S_{3}\)=\(\{\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{5}\} \)的秩依次为\(3,3,4,\)则\(S={\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{5}-\alpha_{4} }\)的秩为______ 。

设`alpha_{1}， alpha_{2}，alpha_{3}`均为3维列向量，记矩阵(A=egin{bmatrix}alpha_{1} ，& alpha_{2} ，& alpha_{3}end{bmatrix})，(B=egin{bmatrix}alpha_{1}+alpha_{2}+alpha_{3} ， & alpha_{1}+2alpha_{2}+4alpha_{3} ， & alpha_{1}+3alpha_{2}+9alpha_{3} end{bmatrix}，) 如果(egin{vmatrix} A end{vmatrix}=1，)那么(egin{vmatrix} B end{vmatrix})=______

设向量组(alpha_{1}=(1,2,-1,0)^T,alpha_{2}=(1,1,0,2)^T,alpha_{3}=(2,1,1,a)^T,)若由(alpha_{1},alpha_{2},alpha_{3})生成的向量空间的维数为2(,)则(a)=______

设向量组(alpha_{1}=(1,2,-1,0)^T,alpha_{2}=(1,1,0,2)^T,alpha_{3}=(2,1,1,a)^T,)若由(alpha_{1},alpha_{2},alpha_{3})生成的向量空间的维数为2(,)则(a)=______

已知\(\alpha_{1}=(1,2,-1,1)^T,\alpha_{2}=(2,0,t,0)^T,\alpha_{3}=(0,-4,5,2)^T\)的秩为2\(,\)则\(t\)=______ 。