Finding Least Squares Solution Review and Others

Rotation Matrix

The rotation matrix
\displaystyle  R=\begin{pmatrix}  \cos\theta & -\sin\theta\\  \sin\theta & \cos\theta  \end{pmatrix}
rotates points in the xy-plane counterclockwise through an angle \theta about the origin.

For example rotating the vector (1,0) 45 degrees counterclockwise gives us:
\displaystyle  \begin{pmatrix}  \cos 45^\circ & -\sin 45^\circ\\  \sin 45^\circ & \cos 45^\circ  \end{pmatrix}  \begin{pmatrix}  1\\  0  \end{pmatrix}  =  \begin{pmatrix}  \frac{\sqrt{2}}{2}\\  \frac{\sqrt{2}}{2}  \end{pmatrix}.

Finding Least Squares Solution

Given Ax=b (inconsistent system), solve
\displaystyle A^TAx=A^Tb instead to get a least squares solution of the original equation.

Projection

If we know a least squares solution \mathbf{u} of A\mathbf{x}=\mathbf{b}, we can find the projection \mathbf{p} of \mathbf{b} onto the column space of A by \displaystyle \mathbf{p}=A\mathbf{u}.

Dimension Theorem for Matrices (Also known as Rank-Nullity Theorem)

If A is a matrix with n columns, then \displaystyle rank(A)+nullity(A)=n.

(rank(A)=number of pivot columns,

nullity(A)=number of non-pivot columns.)

Linear Independence and the Wronskian
A set of vector functions \vec{f_1}(x), \dots, \vec{f_n}(x) from \mathbb{R} to \mathbb{R}^n is linearly independent in the interval (\alpha,\beta) if \displaystyle W[\vec{f_1}(x),\dots,\vec{f_n}(x)]\neq 0 for at least one value of x in the interval (\alpha,\beta).

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