Subsection 2.9 Question 9
Question 9.
Let \(A \in \mathbb R^{m \times n} \text{.}\)
ALWAYS/SOMETIMES/NEVER: The row space of \(A \) is orthogonal to the null space of \(A \text{.}\)
Answer
1 Solution
2 Relevance
3 Resources
ALWAYS
Now prove it!
Vector \(u \) is in the row space of \(A \) if and only if there exists a vector \(x \) such that \(u = A^T x \text{.}\)
Vector \(v \) is in the null space of \(A \) if and only if \(A v = 0 \text{.}\)
Two vector spaces are orthogonal if and only if all vectors in one space are orthogonal to all vectors in the other space.
Let \(u \) be an arbitrary vector in the row space of \(A \) and \(v \) an arbitrary vector in the null space of \(A \text{.}\) Then
\begin{equation*}
\begin{array}{l}
u^T v \\
~~~ = ~~~~ \lt \mbox{ since } u
\mbox{ is in the row space of } A
\mbox{ there exists } x
\mbox{ such that } u = A^T x \gt \\
( A^T x )^T v \\
~~~ = ~~~~ \lt ( A B )^T = B^T A^T \gt \\
x^T (A^T)^T v \\
~~~ = ~~~~ \lt (A^T)^T = A\gt \\
x^T A v \\
~~~ = ~~~~ \lt v \mbox{ is in the null space of } A
\gt \\
x^T 0 ~~~~~~~~\mbox{(} 0
\mbox{ is the zero vector of appropriate size) }
\\
~~~ = ~~~~ \\
0
\end{array}
\end{equation*}
Relevance
This question checks whether you know how to prove assertions about sets.
Resources