Activity 6.2.2

Suppose that $\wvec_1=\threevec10{-2}$ and $\wvec_2=\threevec11{-1}$ form a basis for $W$, a two-dimensional subspace of $\real^3$. We will find a description of the orthogonal complement $W^\perp$.

1. Suppose that the vector $\xvec$ is orthogonal to $\wvec_1$. If we write $\xvec=\threevec{x_1}{x_2}{x_3}$, use the fact that $\wvec_1\cdot\xvec = 0$ to write a linear equation for $x_1$, $x_2$, and $x_3$.

2. Suppose that $\xvec$ is also orthogonal to $\wvec_2$. In the same way, write a linear equation for $x_1$, $x_2$, and $x_3$ that arises from the fact that $\wvec_2\cdot\xvec = 0$.

3. If $\xvec$ is orthogonal to both $\wvec_1$ and $\wvec_2$, these two equations give us a linear system $B\xvec=\zerovec$ for some matrix $B$. Identify the matrix $B$ and write a parametric description of the solution space to the equation $B\xvec = \zerovec$.

4. Since $\wvec_1$ and $\wvec_2$ form a basis for the two-dimensional subspace $W$, any vector $\wvec$ in $W$ can be written as a linear combination $$\wvec = c_1\wvec_1 + c_2\wvec_2\text{.}$$ If $\xvec$ is orthogonal to both $\wvec_1$ and $\wvec_2$, use the distributive property of dot products to explain why $\xvec$ is orthogonal to $\wvec$.

5. Give a basis for the orthogonal complement $W^\perp$ and state the dimension $\dim W^\perp$.

6. Describe $(W^\perp)^\perp$, the orthogonal complement of $W^\perp$.