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Let-W-be-the-subspace-of-R-4-generated-by-vector-1-2-5-3-2-3-1-4-3-8-3-5-find-the-basis-and-dimension-of-W-

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Question Number 56899 by Tawa1 last updated on 26/Mar/19
Let W be the subspace of R^4 generated by vector (1, − 2, 5, − 3), (2, 3, 1, − 4), (3, 8, − 3, − 5) find the basis and dimension of W.
$$\mathrm{Let}\:\:\mathrm{W}\:\:\mathrm{be}\:\mathrm{the}\:\mathrm{subspace}\:\mathrm{of}\:\:\mathbb{R}^{\mathrm{4}} \:\:\mathrm{generated}\:\mathrm{by}\:\mathrm{vector}\: \\ $$$$\left(\mathrm{1},\:−\:\mathrm{2},\:\mathrm{5},\:−\:\mathrm{3}\right),\:\:\:\:\left(\mathrm{2},\:\mathrm{3},\:\mathrm{1},\:−\:\mathrm{4}\right),\:\:\:\left(\mathrm{3},\:\mathrm{8},\:−\:\mathrm{3},\:−\:\mathrm{5}\right)\:\:\mathrm{find}\:\mathrm{the} \\ $$$$\mathrm{basis}\:\mathrm{and}\:\mathrm{dimension}\:\mathrm{of}\:\:\mathrm{W}. \\ $$
Answered by kaivan.ahmadi last updated on 26/Mar/19
determinant (((1 −2 5 −3)),((2 3 1 −4 )),((3 8 −3 −5)))→_(−3R_1 +R_3 ) ^(−2R_1 +R_2 ) determinant (((1 −2 5 −3)),((0 7 −9 2)),((0 14 −18 4))) →^(R_2 /7) determinant (((1 −2 5 −3)),((0 1 ((−9)/7) (2/7))),((0 14 −18 4)))→_(−14R_2 +R_3 ) ^(2R_2 +R_1 ) determinant (((1 0 ((17)/7) ((−17)/7))),((0 1 ((−9)/7) (2/7))),((0 0 0 0)))⇒ dim(W)=2=number of nonzero rows and the set of nonzero rows as vector is a basis for W
$$\begin{vmatrix}{\mathrm{1}\:\:\:\:\:\:\:−\mathrm{2}\:\:\:\:\:\:\:\:\:\mathrm{5}\:\:\:\:\:\:−\mathrm{3}}\\{\mathrm{2}\:\:\:\:\:\:\:\:\:\:\:\mathrm{3}\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:−\mathrm{4}\:\:}\\{\mathrm{3}\:\:\:\:\:\:\:\:\:\:\:\mathrm{8}\:\:\:\:\:−\mathrm{3}\:\:\:\:\:\:\:−\mathrm{5}}\end{vmatrix}\underset{−\mathrm{3}{R}_{\mathrm{1}} +{R}_{\mathrm{3}} } {\overset{−\mathrm{2}{R}_{\mathrm{1}} +{R}_{\mathrm{2}} } {\rightarrow}}\begin{vmatrix}{\mathrm{1}\:\:\:\:\:\:−\mathrm{2}\:\:\:\:\:\:\:\:\:\mathrm{5}\:\:\:\:\:\:\:\:\:−\mathrm{3}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\mathrm{7}\:\:\:\:\:\:\:\:−\mathrm{9}\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}}\\{\mathrm{0}\:\:\:\:\:\:\:\mathrm{14}\:\:\:\:\:\:−\mathrm{18}\:\:\:\:\:\:\:\:\:\:\mathrm{4}}\end{vmatrix} \\ $$$$\overset{\frac{{R}_{\mathrm{2}} }{\mathrm{7}}} {\rightarrow}\begin{vmatrix}{\mathrm{1}\:\:\:\:\:−\mathrm{2}\:\:\:\:\:\:\:\:\mathrm{5}\:\:\:\:\:\:\:\:\:\:−\mathrm{3}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\frac{−\mathrm{9}}{\mathrm{7}}\:\:\:\:\:\:\:\:\frac{\mathrm{2}}{\mathrm{7}}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\mathrm{14}\:\:\:\:\:\:\:−\mathrm{18}\:\:\:\:\:\:\:\mathrm{4}}\end{vmatrix}\underset{−\mathrm{14}{R}_{\mathrm{2}} +{R}_{\mathrm{3}} } {\overset{\mathrm{2}{R}_{\mathrm{2}} +{R}_{\mathrm{1}} } {\rightarrow}} \\ $$$$\begin{vmatrix}{\mathrm{1}\:\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{17}}{\mathrm{7}}\:\:\:\:\:\:\:\:\frac{−\mathrm{17}}{\mathrm{7}}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\mathrm{1}\:\:\:\:\:\:\:\:\:\:\frac{−\mathrm{9}}{\mathrm{7}}\:\:\:\:\:\:\:\:\:\frac{\mathrm{2}}{\mathrm{7}}}\\{\mathrm{0}\:\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{0}}\end{vmatrix}\Rightarrow \\ $$$${dim}\left({W}\right)=\mathrm{2}={number}\:{of}\:{nonzero}\:{rows} \\ $$$${and}\:{the}\:{set}\:{of}\:{nonzero}\:{rows}\:{as}\:{vector}\:{is}\:{a}\:{basis}\:{for}\:{W} \\ $$$$ \\ $$$$ \\ $$
Commented by Tawa1 last updated on 29/Mar/19
God bless you sir. I appreciate
$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}.\:\mathrm{I}\:\mathrm{appreciate}\: \\ $$

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