Paper 1, Section II, 5C5 \mathrm{C}

Vectors and Matrices | Part IA, 2013

Let x\mathbf{x} and y\mathbf{y} 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. Under what condition does equality hold in the Cauchy-Schwarz inequality?

Let x,y,z\mathbf{x}, \mathbf{y}, \mathbf{z} be unit vectors in R3\mathbb{R}^{3}. Let

S=xy+yz+zxS=\mathbf{x} \cdot \mathbf{y}+\mathbf{y} \cdot \mathbf{z}+\mathbf{z} \cdot \mathbf{x}

Show that for any fixed, linearly independent vectors x\mathbf{x} and y\mathbf{y}, the minimum of SS over z\mathbf{z} is attained when z=λ(x+y)\mathbf{z}=\lambda(\mathbf{x}+\mathbf{y}) for some λR\lambda \in \mathbb{R}, and that for this value of λ\lambda we have

(i) λ12\lambda \leqslant-\frac{1}{2} (for any choice of x\mathbf{x} and y)\left.\mathbf{y}\right);

(ii) λ=1\lambda=-1 and S=32S=-\frac{3}{2} in the case where xy=cos2π3\mathbf{x} \cdot \mathbf{y}=\cos \frac{2 \pi}{3}.

Typos? Please submit corrections to this page on GitHub.