Question #206221

show that the set {(1,2,3),(0,1,2),(0,0,1) of vectors generates or span R^3


1
Expert's answer
2021-06-15T04:55:36-0400

We can set up a matrix and use Gaussian elimination to figure out the dimension of the space they span.


(123012001)\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \\ \end{pmatrix}

R1=R12R2R_1=R_1-2R_2


(101012001)\begin{pmatrix} 1 & 0 & -1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \\ \end{pmatrix}

R1=R1+R3R_1=R_1+R_3


(100012001)\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \\ \end{pmatrix}

R2=R22R3R_2=R_2-2R_3


(100010001)\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}

Since the rank of the matrix is 3, then the set {(1,2,3),(0,1,2),(0,0,1) of vectors generates or span R3.R^3.



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