Let W be the subspace of 𝐑^5 spanned by u=(1,2,3,-1,2) and v=(2,4,7,2,-1) . Find a basis of the

step 1 We're given W, which is a subspace of R5, which is spanned by vectors U with components 1, 2, 2,

Let W be the subspace of 𝐑^5 spanned by u=(1,2,3,-1,2) and...

Let $W$ be the subspace of $\mathbf{R}^{5}$ spanned by $u=(1,2,3,-1,2)$ and $v=(2,4,7,2,-1) .$ Find a basis of the orthogonal complement $W^{\perp}$ of $W$ We seck all vectors $w=(x, y, z, s, t)$ such that \[\begin{array}{l}\langle w, u\rangle=x+2 y+3 z-s+2 t=0 \\ \langle w, v\rangle=2 x+4 y+7 z+2 s-t=0\end{array}\] Eliminating $x$ from the second equation, we find the equivalent system \[\begin{array}{r}x+2 y+3 z-s+2 t=0 \\z+4 s-5 t=0\end{array}\] The free variables are $y, s,$ and $t .$ Therefore, (1) $\operatorname{Set} y=-1, s=0, t=0$ to obtain the solution $w_{1}=(2,-1,0,0,0)$ (2) $\operatorname{Set} y=0, s=1, t=0$ to find the solution $w_{2}=(13,0,-4,1,0)$ (3) $\operatorname{Set} y=0, s=0, t=1$ to obtain the solution $w_{3}=(-17,0,5,0,1)$ The set $\left\{w_{1}, w_{2}, w_{3}\right\}$ is a basis of $W^{\perp}$

Let $W$ be the subspace of $\mathbf{R}^{5}$ spanned by $u=(1,2,3,-1,2)$ and $v=(2,4,7,2,-1) .$ Find a basis of the orthogonal complement $W^{\perp}$ of $W$ We seck all vectors $w=(x, y, z, s, t)$ such that
\[\begin{array}{l}\langle w, u\rangle=x+2 y+3 z-s+2 t=0 \\
\langle w, v\rangle=2 x+4 y+7 z+2 s-t=0\end{array}\]
Eliminating $x$ from the second equation, we find the equivalent system
\[\begin{array}{r}x+2 y+3 z-s+2 t=0 \\z+4 s-5 t=0\end{array}\]
The free variables are $y, s,$ and $t .$ Therefore,
(1) $\operatorname{Set} y=-1, s=0, t=0$ to obtain the solution $w_{1}=(2,-1,0,0,0)$
(2) $\operatorname{Set} y=0, s=1, t=0$ to find the solution $w_{2}=(13,0,-4,1,0)$
(3) $\operatorname{Set} y=0, s=0, t=1$ to obtain the solution $w_{3}=(-17,0,5,0,1)$
The set $\left\{w_{1}, w_{2}, w_{3}\right\}$ is a basis of $W^{\perp}$

Key Concepts

System of Linear Equations and Parametrization

Solving a system of linear equations involves finding all vectors that satisfy a set of linear constraints. When the system has free variables, the solutions can be expressed in a parametric form, which typically leads to identifying a basis for the solution space. This technique is essential for characterizing subspaces like the orthogonal complement by parameterizing all vectors that meet the given conditions.

Orthogonal Complement

The orthogonal complement of a subspace W in an inner product space is the set of all vectors that are orthogonal to every vector in W. This concept is significant because it allows the decomposition of a vector space into mutually perpendicular subspaces, which is key in solving problems related to projections and least squares approximations.

Inner Product

The inner product, often instantiated as the dot product in Euclidean spaces, is a function that takes two vectors and returns a scalar. It provides a way to define geometric concepts such as length, angle, and orthogonality in vector spaces, and it is fundamental in constructing orthogonal complements.

Subspace

A subspace is a subset of a vector space that is itself a vector space under the same operations of addition and scalar multiplication. In linear algebra, subspaces are fundamental because they provide a framework for discussing solutions to linear systems and understanding the structure of vector spaces.

Basis

A basis of a vector space (or a subspace) is a set of linearly independent vectors that span the entire space. This means every vector in the space can be uniquely expressed as a linear combination of the basis vectors. The concept of a basis is crucial for understanding dimensions and for performing coordinate changes.

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