Linear Algebra · Lesson 08

Chapter practice

Bring the chapter together: lengths and angles, orthogonal bases and matrices, complements, inner products, projections, Gram–Schmidt and least squares. No worked example sits above the answers.

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u \cdot v = i \sum u_{i} v_{i}

⟨ u, v ⟩

Q^{T} Q = I

u ⊥ v

∥ v ∥^{2} = ⟨ v, v ⟩

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Intuition

Bring the chapter together: lengths and angles, orthogonal bases and matrices, complements, inner products, projections, Gram–Schmidt and least squares. No worked example sits above the answers.

A carpenter checks a frame with a set square, a tape and a level. The dot product is all three at once.

Before you practise

Decide first what is asked. An angle or a length: dot products. Coordinates in an orthogonal basis: one dot product each. A complement: a null space. The closest point of a subspace: a projection, with an orthogonal basis from Gram–Schmidt if needed. An inconsistent system: the normal equations.

\operatorname{proj}_{\mathbf{u}}\mathbf{x}=\frac{\mathbf{x}\cdot\mathbf{u}}{\mathbf{u}\cdot\mathbf{u}}\,\mathbf{u}

Reminders

$\cos\theta=\dfrac{\mathbf{u}\cdot\mathbf{v}}{\|\mathbf{u}\|\,\|\mathbf{v}\|}$ ; orthogonal means dot product $0$ .
$Q^{T}Q=I$ for an orthogonal matrix, and it keeps lengths.
$\dim W+\dim W^{\perp}=n$ and $\operatorname{Nul}A=(\operatorname{Row}A)^{\perp}$ .
$A^{T}A\hat{\mathbf{x}}=A^{T}\mathbf{b}$ .

Applications