Algebra and Geometry | Part IA, 2003

Let x=(x1,x2,,xn)\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right) and y=(y1,y2,,yn)\mathbf{y}=\left(y_{1}, y_{2}, \ldots, y_{n}\right) be non-zero vectors in Rn\mathbb{R}^{n}. What is meant by saying that x\mathbf{x} and y\mathbf{y} are linearly independent? What is the dimension of the subspace of Rn\mathbb{R}^{n} spanned by x\mathbf{x} and y\mathbf{y} if they are (1) linearly independent, (2) linearly dependent?

Define the scalar product xy\mathbf{x} \cdot \mathbf{y} for x,yRn\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}. Define the corresponding norm x\|\mathbf{x}\| of xRn\mathbf{x} \in \mathbb{R}^{n}. State and prove the Cauchy-Schwarz inequality, and deduce the triangle inequality.

By means of a sketch, give a geometric interpretation of the scalar product xy\mathbf{x} \cdot \mathbf{y} in the case n=3n=3, relating the value of xy\mathbf{x} \cdot \mathbf{y} to the angle α\alpha between the directions of x\mathbf{x} and y\mathbf{y}.

What is a unit vector? Let u,v,w\mathbf{u}, \mathbf{v}, \mathbf{w} be unit vectors in R3\mathbb{R}^{3}. Let

S=uv+vw+wuS=\mathbf{u} \cdot \mathbf{v}+\mathbf{v} \cdot \mathbf{w}+\mathbf{w} \cdot \mathbf{u}

Show that

(i) for any fixed, linearly independent u\mathbf{u} and v\mathbf{v}, the minimum of SS over w\mathbf{w} is attained when w=λ(u+v)\mathbf{w}=\lambda(\mathbf{u}+\mathbf{v}) for some λR\lambda \in \mathbb{R};

(ii) λ12\lambda \leqslant-\frac{1}{2} in all cases;

(iii) λ=1\lambda=-1 and S=3/2S=-3 / 2 in the case where uv=cos(2π/3)\mathbf{u} \cdot \mathbf{v}=\cos (2 \pi / 3).

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