正交补

Introduction

Definition of the Orthogonal Complement

Computing Orthogonal Complements

Row rank and column rank

Let $W$ be a $2$-dimensional subspace of $\mathbb{R}^4$. What is $\dim(W^\perp)$?

For a matrix $A$, the equation $A^T y = d$ is solvable exactly when $d$ lies in which of the four fundamental subspaces of $A$?

In two words: for a subspace $W$ of $\mathbb{R}^n$, the set of all vectors that are orthogonal to every vector in $W$ is called the orthogonal ______ of $W$.

Let $W$ be a plane through the origin in $\mathbb{R}^3$ (so $\dim W = 2$). What is $\dim(W^\perp)$? Enter a number.

The system

has no solution. Multiply the equations by $y_1, y_2, y_3$ so that the weighted left sides cancel identically: if the system were consistent, the weighted right sides would then have to satisfy $y_1 + y_2 + y_3 = 0$. Take $y_1 = y_2 = 1$; enter $y_3$ as a number.

For any matrix $A$, the orthogonal complement $\operatorname{Nul}(A)^\perp$ equals which fundamental subspace?

Let $W$ be the line through the origin spanned by $(1,1,1)$ in $\mathbb{R}^3$. Enter $\dim(W^\perp)$ as a number.