Find the orthogonal projection of 𝐯=(1,2,-1,2)^T onto the following subspaces: (a) the span of (

### Part (a): Orthogonal Projection onto the Span of Two Vectorsu000Au000A**Step 1:** Define the vectors

Find the orthogonal projection of 𝐯=(1,2,-1,2)^T onto the...

Find the orthogonal projection of $\mathbf{v}=(1,2,-1,2)^T$ onto the following subspaces: (a) the span of $\left(\begin{array}{r}1 \\ -1 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{r}2 \\ 1 \\ 0 \\ -1\end{array}\right)$; (b) the image of the matrix $\left(\begin{array}{rr}1 & 2 \\ -1 & 1 \\ 0 & 3 \\ -1 & 1\end{array}\right)$; (c) the kernel of the matrix $\left(\begin{array}{rrrr}1 & -1 & 0 & 1 \\ -2 & 1 & 1 & 0\end{array}\right)$; (d) the subspace orthogonal to $\mathbf{a}=(1,-1,0,1)^T$. Warning. Make sure you have an orthogonal basis before applying formula (4.42)!

Find the orthogonal projection of $\mathbf{v}=(1,2,-1,2)^T$ onto the following subspaces:
(a) the span of $\left(\begin{array}{r}1 \\ -1 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{r}2 \\ 1 \\ 0 \\ -1\end{array}\right)$;
(b) the image of the matrix $\left(\begin{array}{rr}1 & 2 \\ -1 & 1 \\ 0 & 3 \\ -1 & 1\end{array}\right)$;
(c) the kernel of the matrix $\left(\begin{array}{rrrr}1 & -1 & 0 & 1 \\ -2 & 1 & 1 & 0\end{array}\right)$;
(d) the subspace orthogonal to $\mathbf{a}=(1,-1,0,1)^T$. Warning. Make sure you have an orthogonal basis before applying formula (4.42)!

Key Concepts

Orthogonal Basis

An orthogonal basis is a set of basis vectors for a subspace that are mutually perpendicular. Using an orthogonal (or orthonormal) basis simplifies the computation of orthogonal projections since the coefficients in the projection formula are obtained directly through the dot product without needing to solve linear systems.

Kernel

The kernel (or nullspace) of a matrix is the set of all vectors that the matrix maps to the zero vector. It constitutes a subspace of the domain of the matrix and plays a key role in understanding solutions to homogeneous linear equations, as well as in projection problems where one is interested in components orthogonal to certain directions.

Image

The image of a matrix (or linear transformation) is the set of all vectors that can be obtained by applying the matrix to vectors from its domain. It is a subspace of the codomain, and understanding this concept is important when projecting a vector onto the output space of a linear transformation.

Orthogonal Complement

The orthogonal complement of a subspace consists of all vectors that are orthogonal to every vector in the subspace. This concept is important when projecting onto or away from a given subspace because it allows decomposition of a vector into parts that lie within the subspace and its complement.

Subspace

A subspace is a subset of a vector space that is itself a vector space under the same operations. Subspaces can be described in various ways such as the span of a set of vectors, the image of a matrix, or the kernel of a matrix, and they serve as the context in which projection operations are performed.

Orthogonal Projection

The orthogonal projection of a vector onto a subspace is the component of the vector that lies in the subspace, found by ‘dropping a perpendicular’ from the vector to the subspace. This concept is fundamental in linear algebra and is used to minimize the distance between a vector and the subspace, often using an orthogonal or orthonormal basis of the subspace to simplify calculations.

Span

The span of a set of vectors is the set of all linear combinations of those vectors. It represents the smallest subspace that contains the given vectors, and is a common way to define the subspace onto which projections are computed.

Please give Ace some feedback