Tagged: row vector

12/26/2017

The Set of Vectors Perpendicular to a Given Vector is a Subspace

Problem 659

Fix the row vector $\mathbf{b} = \begin{bmatrix} -1 & 3 & -1 \end{bmatrix}$, and let $\R^3$ be the vector space of $3 \times 1$ column vectors. Define
\[W = \{ \mathbf{v} \in \R^3 \mid \mathbf{b} \mathbf{v} = 0 \}.\] Prove that $W$ is a vector subspace of $\R^3$.

Add to solve later

10/05/2017

The Column Vectors of Every $3\times 5$ Matrix Are Linearly Dependent

Problem 580

(a) Prove that the column vectors of every $3\times 5$ matrix $A$ are linearly dependent.

(b) Prove that the row vectors of every $5\times 3$ matrix $B$ are linearly dependent.

Add to solve later

10/07/2016

Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose)

Problem 136

Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.

Add to solve later