Algebra and Geometry | Part IA, 2001

Define what is meant by the statement that the vectors x1,,xnRm\mathbf{x}_{1}, \ldots, \mathbf{x}_{n} \in \mathbb{R}^{m} are linearly independent. Determine whether the following vectors x1,x2,x3R3\mathbf{x}_{1}, \mathbf{x}_{2}, \mathbf{x}_{3} \in \mathbb{R}^{3} are linearly independent and justify your answer.

x1=(132),x2=(240),x3=(104)\mathbf{x}_{1}=\left(\begin{array}{l} 1 \\ 3 \\ 2 \end{array}\right), \quad \mathbf{x}_{2}=\left(\begin{array}{l} 2 \\ 4 \\ 0 \end{array}\right), \quad \mathbf{x}_{3}=\left(\begin{array}{c} -1 \\ 0 \\ 4 \end{array}\right)

For the vectors x,y,z\mathbf{x}, \mathbf{y}, \mathbf{z} taken from a real vector space VV consider the statements A) x,y,z\mathbf{x}, \mathbf{y}, \mathbf{z} are linearly dependent, B) α,β,γR:αx+βy+γz=0\exists \alpha, \beta, \gamma \in \mathbb{R}: \alpha \mathbf{x}+\beta \mathbf{y}+\gamma \mathbf{z}=\mathbf{0}, C) α,β,γR\exists \alpha, \beta, \gamma \in \mathbb{R}, not all =0:αx+βy+γz=0=0: \alpha \mathbf{x}+\beta \mathbf{y}+\gamma \mathbf{z}=\mathbf{0}, D) α,βR\exists \alpha, \beta \in \mathbb{R}, not both =0:z=αx+βy=0: \mathbf{z}=\alpha \mathbf{x}+\beta \mathbf{y}, E) α,βR:z=αx+βy\exists \alpha, \beta \in \mathbb{R}: \mathbf{z}=\alpha \mathbf{x}+\beta \mathbf{y}, F) \nexists basis of VV that contains all 3 vectors x,y,z\mathbf{x}, \mathbf{y}, \mathbf{z}.

State if the following implications are true or false (no justification is required): i) AB\mathrm{A} \Rightarrow \mathrm{B}, vi) BA\mathrm{B} \Rightarrow \mathrm{A}, ii) AC\mathrm{A} \Rightarrow \mathrm{C}, vii) CA\mathrm{C} \Rightarrow \mathrm{A}, iii) AD\mathrm{A} \Rightarrow \mathrm{D}, viii) DA\mathrm{D} \Rightarrow \mathrm{A}, iv) AE\mathrm{A} \Rightarrow \mathrm{E}, ix) EA\mathrm{E} \Rightarrow \mathrm{A}, v) AF\mathrm{A} \Rightarrow \mathrm{F}, x) FA\quad \mathrm{F} \Rightarrow \mathrm{A}.

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