Use the projection matrix method of Exercise 4.4.10 to find...

Use the projection matrix method of Exercise 4.4.10 to find the orthogonal projection of $\mathbf{v}=(1,0,0,0)^T$ onto the image of the following matrices: (a) $\left(\begin{array}{r}5 \\ -5 \\ -7 \\ 1\end{array}\right)$, (b) $\left(\begin{array}{rr}1 & 0 \\ -1 & 2 \\ 0 & -1 \\ 1 & 2\end{array}\right)$, (c) $\left(\begin{array}{rr}2 & -1 \\ -3 & 1 \\ 1 & -2 \\ 1 & 2\end{array}\right)$ (d) $\left(\begin{array}{rrr}0 & 1 & -1 \\ 0 & -1 & 2 \\ 1 & 1 & 1 \\ -2 & -1 & 0\end{array}\right)$

Use the projection matrix method of Exercise 4.4.10 to find the orthogonal projection of $\mathbf{v}=(1,0,0,0)^T$ onto the image of the following matrices:
(a) $\left(\begin{array}{r}5 \\ -5 \\ -7 \\ 1\end{array}\right)$,
(b) $\left(\begin{array}{rr}1 & 0 \\ -1 & 2 \\ 0 & -1 \\ 1 & 2\end{array}\right)$,
(c) $\left(\begin{array}{rr}2 & -1 \\ -3 & 1 \\ 1 & -2 \\ 1 & 2\end{array}\right)$
(d) $\left(\begin{array}{rrr}0 & 1 & -1 \\ 0 & -1 & 2 \\ 1 & 1 & 1 \\ -2 & -1 & 0\end{array}\right)$

Key Concepts

Orthogonal Projection

Orthogonal projection is the process of finding the closest point in a subspace to a given vector. It minimizes the distance between the given vector and any vector in the subspace by ensuring that the error (the difference between the given vector and its projection) is orthogonal to every vector in the subspace.

Projection Matrix Method

The projection matrix method involves creating a matrix that, when multiplied by any vector, yields its orthogonal projection onto a specified subspace. In the context of a subspace represented by the columns of a matrix A, the projection matrix is often given by P = A (A?A)?¹ A?, assuming that A has full column rank. This method systematically derives the projection by leveraging the properties of orthogonality and least squares.

Column Space

The column space (or image) of a matrix is the set of all possible linear combinations of its columns. It represents the subspace onto which a given vector is projected. Understanding the column space is crucial because the projection of a vector onto this space is the component of the vector that lies within the span of the matrix’s columns.

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