nonzero solution | Problems in Mathematics

12/25/2017

Show that the Given 2 by 2 Matrix is Singular

Consider the matrix $M = \begin{bmatrix} 1 & 4 \\ 3 & 12 \end{bmatrix}$.

(a) Show that $M$ is singular.

(b) Find a non-zero vector $\mathbf{v}$ such that $M \mathbf{v} = \mathbf{0}$, where $\mathbf{0}$ is the $2$-dimensional zero vector.

09/29/2017

Let $V$ be a subspace of $\R^n$.
Suppose that
\[S=\{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m\}\] is a spanning set for $V$.

Prove that any set of $m+1$ or more vectors in $V$ is linearly dependent.