Determine if following subsets of R^2 are subspaces of R^2. If the subset is a subspace show that it is closed under vector addition and scalar multiplication. If the subset is not a subspace show why, indicating property that fails.(adsbygoogle = window.adsbygoogle || []).push({});

[tex] 1) W=\{ \left (x_1,0)\left| x_1\in\Re\} \newline [/tex]

[tex] 2) W=\{ \left (x_1,0)\left| x_1 > 0\} \newline[/tex]

[tex]3) W=\{ \left (2c,-3c)\left| c \in\Re\} \newline[/tex]

[tex]4) W=\{ \left (x_1,x_2)\left| x_1 > 0, x_2>0\} \newline [/tex]

Answers:

[tex] 1) (x,0) + (y,0)= (x+y,0) \in W \} \newline [/tex]

[tex] c(x,0) = (c(x),0) \in W \} \newline [/tex]

2) Not subspace since x1 can't be 0

[tex] 3) (2c,-3c) + (x_1,x_2)= (2c+x_1,-3c+x_2) \in W \} \newline [/tex]

[tex] x(2c,-3c) = (2c(x),-3c(x)) \in W \} \newline [/tex]

4) Not a subspace since x1 and x2 can't be 0.

Am I on the right track with this? Thanks

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# Another vector problem

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