Quizmaster

Learning objectives

What you will be able to explain

Section 01

Rank and consistency determine whether inverse, least-squares, or null-space descriptions apply.

Guided checkpoint

Match the system to the canonical solution concept.

Source: Sec. 5.1, pp. 28-30

x=(A^TA)^{-1}A^Tb=(1+4)^{-1}(2+10)=12/5

Guided checkpoint

Find the least-squares scalar $x$ for $A=[1,2]^T$ and $b=[2,5]^T$ .

Source: Sec. 5.1, pp. 28-30

A complete eigenbasis is not guaranteed for defective matrices.

Guided checkpoint

For square $A$ .

Source: Sec. 5.2, pp. 30-31

Section 02

The SVD separates orthogonal input/output directions and nonnegative gains.

Guided checkpoint

For $A=U Sigma V^T$ , match the role.

Source: Sec. 5.3, pp. 31-32

Triangular factors make solves and determinants inexpensive; Cholesky has strong assumptions.

Guided checkpoint

Check each claim.

Source: Sec. 5.4, p. 32

The decompositions differ in symmetry, normalization, and where diagonal scaling resides.

Guided checkpoint

Match factorization to its structure.

Source: Secs. 5.4-5.7, pp. 32-33

Section 03

Normalize

a

and

b

as singular vectors; the remaining scale is the product of their norms.

Guided checkpoint

For nonzero vectors $a,b$ , what is the sole singular value of $ab^T$ ?

Source: Sec. 5.3, pp. 31-32

Knowledge check

The quiz now follows the same concepts in scored form. You can return to this lesson from the quiz whenever a gap appears.