Subset of Vectors Perpendicular to Two Vectors is a Subspace
Problem 119
Let $\mathbf{a}$ and $\mathbf{b}$ be fixed vectors in $\R^3$, and let $W$ be the subset of $\R^3$ defined by
\[W=\{\mathbf{x}\in \R^3 \mid \mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0\}.\]
Prove that the subset $W$ is a subspace of $\R^3$.
We prove the following criteria for the subset $W$ to be a subspace of $\R^3$.
(a) The zero vector $\mathbf{0} \in \R^3$ is in $W$.
(b) If $\mathbf{x}, \mathbf{y} \in W$, then $\mathbf{x}+\mathbf{y}\in W$.
(c) If $\mathbf{x} \in W$ and $c\in \R$, then $c\mathbf{x} \in W$.
For (a), note that $\mathbf{a}^{\trans} \mathbf{0}=0$ and $\mathbf{b}^{\trans} \mathbf{0}=0$. Thus the zero vector $\mathbf{0}\in \R^3$ is in $W$.
To check (b), let $\mathbf{x}, \mathbf{y} \in W$. Then we have the following relations.
\[\mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0,
\text{ and }
\mathbf{a}^{\trans} \mathbf{y}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{y}=0.\tag{*}\]
To show that $\mathbf{x}+\mathbf{y} \in W$, we need to show that
\[\mathbf{a}^{\trans}(\mathbf{x}+\mathbf{y})=0 \text{ and } \mathbf{b}^{\trans}(\mathbf{x}+\mathbf{y})=0.\]
We first compute
\begin{align*}
\mathbf{a}^{\trans}(\mathbf{x}+\mathbf{y}) &=\mathbf{a}^{\trans}\mathbf{x}+\mathbf{a}^{\trans}\mathbf{y}\\
&= 0+0=0
\end{align*}
by the relations (*).
Similarly, we have
\begin{align*}
\mathbf{b}^{\trans}(\mathbf{x}+\mathbf{y}) &=\mathbf{b}^{\trans}\mathbf{x}+\mathbf{b}^{\trans}\mathbf{y}\\
&= 0+0=0
\end{align*}
by the relations (*).
Thus the vector $\mathbf{x}+\mathbf{y}$ satisfies the defining relations for $W$, hence $\mathbf{x}+\mathbf{y} \in W$.
Finally, to prove (c), let $\mathbf{x} \in W$ and let $c\in \R$.
Since $\mathbf{x} \in W$, we have
\[\mathbf{a}^{\trans} \mathbf{x}=0 \text{ and } \mathbf{b}^{\trans} \mathbf{x}=0.\]
Multiplying by the scalar $c$ from the left, we obtain
\[\mathbf{a}^{\trans} (\mathbf{cx})=0 \text{ and } \mathbf{b}^{\trans} (\mathbf{cx})=0.\]
(Note that since $c$ is a scalar, we can switch the order of the product of $c$ and $\mathbf{a}^{\trans}$. Same for $c$ and $\mathbf{b}^{\trans}$.)
These equalities proves that the vector $c\mathbf{x}$ satisfies the defining relation for $W$. Thus $c\mathbf{x} \in W$.
Therefore the criteria (a)-(c) are all met, and we conclude that $W$ is a subspace of $\R^3$.
Find a Condition that a Vector be a Linear Combination
Let
\[\mathbf{v}=\begin{bmatrix}
a \\
b \\
c
\end{bmatrix}, \qquad \mathbf{v}_1=\begin{bmatrix}
1 \\
2 \\
0
\end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix}
2 \\
-1 \\
2
\end{bmatrix}.\]
Find the necessary and […]
Find the Distance Between Two Vectors if the Lengths and the Dot Product are Given
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\R^n$ such that their length are
\[\|\mathbf{a}\|=\|\mathbf{b}\|=1\]
and the inner product
\[\mathbf{a}\cdot \mathbf{b}=\mathbf{a}^{\trans}\mathbf{b}=-\frac{1}{2}.\]
Then determine the length $\|\mathbf{a}-\mathbf{b}\|$.
(Note […]
Sherman-Woodbery Formula for the Inverse Matrix
Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\R^n$, and let $I$ be the $n \times n$ identity matrix. Suppose that the inner product of $\mathbf{u}$ and $\mathbf{v}$ satisfies
\[\mathbf{v}^{\trans}\mathbf{u}\neq -1.\]
Define the matrix […]
Orthogonality of Eigenvectors of a Symmetric Matrix Corresponding to Distinct Eigenvalues
Suppose that a real symmetric matrix $A$ has two distinct eigenvalues $\alpha$ and $\beta$.
Show that any eigenvector corresponding to $\alpha$ is orthogonal to any eigenvector corresponding to $\beta$.
(Nagoya University, Linear Algebra Final Exam Problem)
Hint.
Two […]
Rotation Matrix in Space and its Determinant and Eigenvalues
For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by
\[A=\begin{bmatrix}
\cos\theta & -\sin\theta & 0 \\
\sin\theta &\cos\theta &0 \\
0 & 0 & 1
\end{bmatrix}.\]
(a) Find the determinant of the matrix $A$.
(b) Show that $A$ is an […]
Inner Product, Norm, and Orthogonal Vectors
Let $\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3$ are vectors in $\R^n$. Suppose that vectors $\mathbf{u}_1$, $\mathbf{u}_2$ are orthogonal and the norm of $\mathbf{u}_2$ is $4$ and $\mathbf{u}_2^{\trans}\mathbf{u}_3=7$. Find the value of the real number $a$ in […]
Express the vector $\mathbf{b}=\begin{bmatrix} 2 \\ 13 \\ 6 \end{bmatrix}$ as a linear combination of the vectors \[\mathbf{v}_1=\begin{bmatrix} 1 \\...