Question

Suppose that А is an mxn matrix with independent columns and the equation Añ = is inconsistent. Then the following statements are true. A. The least squares solution to Ax = 6 is given by î = (A” A) A 5 B. We can reduce the least squares solution Î = (A” A)-'A” as follows. î = (A” A)'AT = Â = A-'(AT)-'A" 6 = This calculation follows since when matrices

Answer 1

Correct answer is true.

Explanation:-

For an inconsistent system Ax = b, where a solution does not exist, the best we can do is to find an x that
makes Ax as close as possible to b. By the Best Approximation theorem, we have:
Definition. If A is m × n and b ∈ R
n, a least-squares solution of Ax = b is a vector xˆ ∈ R
n such that
kb − Axˆk ≤ kb − Axk
for all x ∈ R
n.
The adjective “least-squares” arises from the fact that kb − Axˆk is the square root of a sum of squares.
The most important aspect of the least-squares problem is that no matter what x we select, the vector Ax
will necessarily be in Col A. So we seek a vector x that makes Ax the closest point in Col A to b.
Given A and b, we apply the Best Approximation theorem to the subspace Col A. Let
bˆ = projCol Ab.
Because bˆ is in the column space of A, the equation Ax = bˆ is consistent, and there is an xˆ ∈ R
n such that
Axˆ = bˆ. (1)
Since bˆ is the closest point in Col A to b, a vector xˆ is a least-squares solution of Ax = b if and only if xˆ
satisfies (1). Such an xˆ ∈ R
n is a list of weights that will build bˆ out of the columns of A.
Suppose xˆ satisfies (1). By the Orthogonal Decomposition theorem, the projection bˆ has the property
that b − bˆ is orthogonal to Col A, so b − Axˆ is orthogonal to each column of A. If aj is any column of A,
then aj · (b − Axˆ) = 0, and a
T
j
(b − Axˆ) = 0. Since each aj is a row of AT
,
A
T
(b − Axˆ) = 0. (2)
Thus
A
T b − A
T Axˆ = 0
∴ A
T Axˆ = A
T b.
These calculations show that each least-squares solution of Ax = b satisfies the equation
A
T Ax = A
T b. (3)
The matrix equation (3) represents a system of equations called the normal equations for Ax = b. A solution
of (3) is often denoted by xˆ.
Theorem 13. The set of least-squares solutions of Ax = b coincides with the nonempty set of solutions of
the normal equations AT Ax = AT b.
Theorem 14. Let A be an m × n matrix. The following are equivalent:
1. The equation Ax = b has a unique least-squares solution for each b ∈ R
m.
2. The columns of A are linearly independent.
3. The matrix AT A is invertible.
When these statements are true, the least-squares solution xˆ is given by
xˆ = (A^T A)^-1A^T b (4)

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