Answer:
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).
