Linear Algebra

Find all Column Vector $\mathbf{w}$ such that $\mathbf{v}\mathbf{w}=0$ for a Fixed Vector $\mathbf{v}$

Problem 641

Let $\mathbf{v} = \begin{bmatrix} 2 & -5 & -1 \end{bmatrix}$.

Find all $3 \times 1$ column vectors $\mathbf{w}$ such that $\mathbf{v} \mathbf{w} = 0$.

Solution.

Let $\mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix}$.

Then we want
\[\mathbf{v} \mathbf{w} =\begin{bmatrix} 2 & -5 & -1 \end{bmatrix} \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix}=2 w_1 – 5 w_2 – w_3 = 0.\]

Letting $w_1$ and $w_2$ be free variables, we solve $w_3 = 2w_1 – 5w_2$.
Then every solution to the equation $\mathbf{v} \mathbf{w} = 0$ is of the form
\[\mathbf{w} = \begin{bmatrix} w_1 \\ w_2 \\ 2 w_1 – 5 w_2 \end{bmatrix} = w_1 \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix} + w_2 \begin{bmatrix} 0 \\ 1 \\ -5 \end{bmatrix}.\]

Add to solve later

More from my site

Linear Properties of Matrix Multiplication and the Null Space of a Matrix Let $A$ be an $m \times n$ matrix. Let $\calN(A)$ be the null space of $A$. Suppose that $\mathbf{u} \in \calN(A)$ and $\mathbf{v} \in \calN(A)$. Let $\mathbf{w}=3\mathbf{u}-5\mathbf{v}$. Then find $A\mathbf{w}$. Hint. Recall that the null space of an […]
Basis For Subspace Consisting of Matrices Commute With a Given Diagonal Matrix Let $V$ be the vector space of all $3\times 3$ real matrices. Let $A$ be the matrix given below and we define \[W=\{M\in V \mid AM=MA\}.\] That is, $W$ consists of matrices that commute with $A$. Then $W$ is a subspace of $V$. Determine which matrices are in the subspace $W$ […]
If matrix product $AB$ is a square, then is $BA$ a square matrix? Let $A$ and $B$ are matrices such that the matrix product $AB$ is defined and $AB$ is a square matrix. Is it true that the matrix product $BA$ is also defined and $BA$ is a square matrix? If it is true, then prove it. If not, find a […]
Symmetric Matrices and the Product of Two Matrices Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings. (a) The product $AB$ is symmetric if and only if $AB=BA$. (b) If the product $AB$ is a diagonal matrix, then $AB=BA$. Hint. A matrix $A$ is called symmetric if $A=A^{\trans}$. In […]
If the Matrix Product $AB=0$, then is $BA=0$ as Well? Let $A$ and $B$ be $n\times n$ matrices. Suppose that the matrix product $AB=O$, where $O$ is the $n\times n$ zero matrix. Is it true that the matrix product with opposite order $BA$ is also the zero matrix? If so, give a proof. If not, give a […]
A Matrix Commuting With a Diagonal Matrix with Distinct Entries is Diagonal Let \[D=\begin{bmatrix} d_1 & 0 & \dots & 0 \\ 0 &d_2 & \dots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \dots & d_n \end{bmatrix}\] be a diagonal matrix with distinct diagonal entries: $d_i\neq d_j$ if $i\neq j$. Let $A=(a_{ij})$ be an $n\times n$ matrix […]
$Prove $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ and determine those $\mathbf{x}$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$$ Prove $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ and determine those $\mathbf{x}$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$ For each of the following matrix $A$, prove that $\mathbf{x}^{\trans}A\mathbf{x} \geq 0$ for all vectors $\mathbf{x}$ in $\R^2$. Also, determine those vectors $\mathbf{x}\in \R^2$ such that $\mathbf{x}^{\trans}A\mathbf{x}=0$. (a) $A=\begin{bmatrix} 4 & 2\\ 2& […]
If a Matrix $A$ is Singular, There Exists Nonzero $B$ such that the Product $AB$ is the Zero Matrix Let $A$ be an $n\times n$ singular matrix. Then prove that there exists a nonzero $n\times n$ matrix $B$ such that \[AB=O,\] where $O$ is the $n\times n$ zero matrix. Definition. Recall that an $n \times n$ matrix $A$ is called singular if the […]