Determine if a. H={(x,y)/ y=3x-1} is a subspace of R2 b. H={at+b/ b=8a} is a subspace of P1

Vectors and spaces
asked 2021-02-09
Determine if
\(\displaystyle{a}.{H}={\left\lbrace\frac{{{x},{y}}}{{y}}={3}{x}-{1}\right\rbrace}\) is a subspace of R2
\(\displaystyle{b}.{H}={\left\lbrace{a}{t}+\frac{{b}}{{b}}={8}{a}\right\rbrace}\) is a subspace of P1

Answers (1)


Let V be a vector space and \(S\subset V\) is said to be a subspace of V if the followings holds
\(0\in S\) for u,\(v\in S\) and for a scaler k \(u+kv\in S\).
(a)Here it is given \(\displaystyle{H}={\left\lbrace{\left({x},{y}\right)}:{y}={3}{x}−{1}\right\rbrace}.\) Since (0, 0) is not in H. Thus H is not a subspace of \(R^{2}\).
(b)Here it is given \(\displaystyle{H}={\left\lbrace{a}{t}+{b}:{b}={8}{a}\right\rbrace}.\) Clearly (0, 0) is in H. Let \(at+b,ct+d \in H\). Then \(b=8a\ and\ d=8c\). Now for a scaler k

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