Decompose each of the following vectors with respect to the...

Decompose each of the following vectors with respect to the indicated subspace as $\mathbf{v}=\mathbf{w}+\mathbf{z}$, where $\mathbf{w} \in W, \mathbf{z} \in W^{\perp}$. (a) $\mathbf{v}=\left(\begin{array}{l}1 \\ 2\end{array}\right), W=\operatorname{span}\left\{\left(\begin{array}{r}-3 \\ 1\end{array}\right)\right\}$ (b) $\mathbf{v}=\left(\begin{array}{r}1 \\ 2 \\ -1\end{array}\right), W=\operatorname{span}\left\{\left(\begin{array}{r}-3 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 0 \\ 5\end{array}\right)\right\} ; \quad(c) \mathbf{v}=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right), W=\operatorname{ker}\left(\begin{array}{rrr}1 & 2 & -1 \\ 2 & 0 & 2\end{array}\right)$; (d) $\mathbf{v}=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right), W=\operatorname{img}\left(\begin{array}{rrr}1 & 0 & 1 \\ -2 & -1 & 0 \\ 1 & 3 & -5\end{array}\right) ;(e) \mathbf{v}=\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right), W=\operatorname{ker}\left(\begin{array}{rrrr}1 & 0 & 0 & 2 \\ -2 & -1 & 1 & -3\end{array}\right)$.

Decompose each of the following vectors with respect to the indicated subspace as $\mathbf{v}=\mathbf{w}+\mathbf{z}$, where $\mathbf{w} \in W, \mathbf{z} \in W^{\perp}$. (a) $\mathbf{v}=\left(\begin{array}{l}1 \\ 2\end{array}\right), W=\operatorname{span}\left\{\left(\begin{array}{r}-3 \\ 1\end{array}\right)\right\}$
(b) $\mathbf{v}=\left(\begin{array}{r}1 \\ 2 \\ -1\end{array}\right), W=\operatorname{span}\left\{\left(\begin{array}{r}-3 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 0 \\ 5\end{array}\right)\right\} ; \quad(c) \mathbf{v}=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right), W=\operatorname{ker}\left(\begin{array}{rrr}1 & 2 & -1 \\ 2 & 0 & 2\end{array}\right)$;
(d) $\mathbf{v}=\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right), W=\operatorname{img}\left(\begin{array}{rrr}1 & 0 & 1 \\ -2 & -1 & 0 \\ 1 & 3 & -5\end{array}\right) ;(e) \mathbf{v}=\left(\begin{array}{l}1 \\ 0 \\ 0 \\ 1\end{array}\right), W=\operatorname{ker}\left(\begin{array}{rrrr}1 & 0 & 0 & 2 \\ -2 & -1 & 1 & -3\end{array}\right)$.

Key Concepts

Vector Decomposition

Vector decomposition refers to the process of breaking a vector into two or more components that together sum to the original vector. In the context of subspaces, this often involves expressing a vector as the sum of one component that lies in a given subspace and another that lies in the subspace's orthogonal complement. This unique splitting is fundamental in many linear algebra applications, including least-squares approximation and signal processing.

Subspace

A subspace is a subset of a vector space that is closed under addition and scalar multiplication, meaning any linear combination of vectors in the subspace remains within the subspace. Subspaces can be defined by spanning sets, the kernel of a linear transformation, or the image of a matrix, and they provide the framework necessary for understanding many linear algebra concepts.

Orthogonal Complement

The orthogonal complement of a subspace is the collection of all vectors in the ambient space that are perpendicular to every vector in the subspace. This concept is crucial for decomposing vectors into parts that lie in a subspace and parts that are completely independent of it, often used in projection and error analysis.

Orthogonal Projection

Orthogonal projection is the process of finding the component of a vector that lies within a given subspace such that the residual (the difference between the original vector and its projection) is orthogonal to the subspace. This technique is essential for many applications, including solving systems of equations in the least-squares sense and for reducing dimensionality in data analysis.

Kernel and Image

The kernel (or null space) of a linear map consists of all vectors that map to the zero vector, while the image (or range) is the set of all vectors that can be obtained as outputs of the linear map. Both concepts are key in linear algebra for understanding the structure and properties of linear transformations, including concepts related to invertibility, dimensionality, and the fundamental theorem of linear algebra.

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