What rank-r approximation exactly reproduces the original picture? You only need to answer the question. Do not include the picture. Hint: Think about the original picture. Do not just answer this question by looking at the gures and guessing.

See solving

Step-by-step explanation:

This notes assumes that the reader understands the following

concepts:

• Linear combination of vectors.

• Linearly independent columns.

• Matrix-vector multiplication forms a linear combination of

the columns of the matrix: Let A ∈ R

m×n and x ∈ R

n.

Partition

A →

a0 a1 · · · an−1

and x =

χ0

χ1

.

.

.

χn−1

then

Ax = χ0a0 + χ1a1 + · · · + χn−1an−1.

• Ax = y only if y is in the column space of A (y ∈ C(A)).

• If A has linearly independent columns and y is not in C(A)

(and even if it is), the vector x that comes closest to solving

Ax ≈ y is given by x = (AT A)

−1AT y. Here (AT A)

−1AT

is known as the pseudo-inverse. In this case the vector

z = Ax = A(AT A)

−1AT y is the projection of y onto the

column space of A, C(A).