Hello, just wondering if my proof is sufficient.(adsbygoogle = window.adsbygoogle || []).push({});

Here is the question from my book:

Show that the following sets of elements inR2 form subspaces:

(a) The set of all (x,y) such that x = y.

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So if we call this set W, then we must show the following:

(i) [tex]0 \in W[/tex]

(ii) if [tex] v,w \in W[/tex], then [tex]v+w \in W[/tex]

(iii) if [tex]c \in R[/tex] and [tex]v \in W[/tex] then [tex]cv \in W[/tex]

Pf:

(i) [tex]0 \in W[/tex] because we can take x = 0 = y

(ii) if [tex]v,w \in W[/tex] then [tex](v,v) \in W[/tex] and [tex](w,w) \in W[/tex] and [tex](v,v) + (w,w) = (v + w, v + w) \in W[/tex] so [tex]v+w \in W[/tex] because v + w = x = y = v + w

(iii) if [tex]c \in R[/tex] and [tex]v \in W[/tex], then [tex](v,v) \in W[/tex] and [tex]c(v,v) = (cv,cv) \in W[/tex] so [tex]cv \in W[/tex] because cv = x = y = cv

Therefore W is a subspace.

Does that look just fine? Thanks.

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# Homework Help: Linear Algebra - Subspaces proof

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