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.

Proof.

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

Note that the column vectors of the matrix $A$ are linearly dependent if the matrix equation
$A\mathbf{x}=\mathbf{0}$ has a nonzero solution $\mathbf{x}\in \R^5$.

The equation is equivalent to a $3\times 5$ homogeneous system.
As there are more variables than equations, the homogeneous system has infinitely many solutions.

In particular, the equation has a nonzero solution $\mathbf{x}$.
Hence the column vectors are linearly dependent.

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

Observe that the row vectors of the matrix $B$ are the column vectors of the transpose $B^{\trans}$. Note that the size of $B^{\trans}$ is $3\times 5$.

In part (a), we showed that the column vectors of any $3\times 5$ matrix are linearly dependent.
It follows that the column vectors of $B^{\trans}$ are linearly dependent.
Hence the row vectors of $B$ are linearly dependent.

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