Activity 6.2.3

This activity illustrates how multiplying a vector by $A^T$ is related to computing dot products with the columns of $A$. You'll develop a better understanding of this relationship if you compute the dot products and matrix products in this activity without using technology.

1. If $B = \begin{bmatrix} 3 & 4 \\ -1 & 2 \\ 0 & -2 \\ \end{bmatrix}$, write the matrix $B^T$.

2. Suppose that $$\vvec_1=\threevec20{-2},\hspace{24pt} \vvec_2=\threevec112,\hspace{24pt} \wvec=\threevec{-2}23\text{.}$$ Find the dot products $\vvec_1\cdot\wvec$ and $\vvec_2\cdot\wvec$.

3. Now write the matrix $A = \begin{bmatrix} \vvec_1 & \vvec_2 \end{bmatrix}$ and its transpose $A^T$. Find the product $A^T\wvec$ and describe how this product computes both dot products $\vvec_1\cdot\wvec$ and $\vvec_2\cdot\wvec$.

4. Suppose that $\xvec$ is a vector that is orthogonal to both $\vvec_1$ and $\vvec_2$. What does this say about the dot products $\vvec_1\cdot\xvec$ and $\vvec_2\cdot\xvec$? What does this say about the product $A^T\xvec$?

5. Use the matrix $A^T$ to give a parametric description of all the vectors $\xvec$ that are orthogonal to $\vvec_1$ and $\vvec_2$.

6. Remember that $\nul(A^T)$, the null space of $A^T$, is the solution set of the equation $A^T\xvec=\zerovec$. If $\xvec$ is a vector in $\nul(A^T)$, explain why $\xvec$ must be orthogonal to both $\vvec_1$ and $\vvec_2$.

7. Remember that $\col(A)$, the column space of $A$, is the set of linear combinations of the columns of $A$. Therefore, any vector in $\col(A)$ can be written as $c_1\vvec_1+c_2\vvec_2$. If $\xvec$ is a vector in $\nul(A^T)$, explain why $\xvec$ is orthogonal to every vector in $\col(A)$.