  Let $F_2$ be the prime field of two elements and let $GL_s:= GL(s, F_2)$ be
the general linear group of rank $s.$ Denote by $\mathscr A$ the Steenrod
algebra over $F_2.$ The (mod-2) Lambda algebra, $\Lambda,$ is one of the tools
to describe those mysterious "Ext-groups". In addition, the $s$-th algebraic
transfer of William Singer \cite{Singer} is also expected to be a useful tool
in the study of them. This transfer is a homomorphism $Tr_s: F_2
\otimes_{GL_s}P_{\mathscr A}(H_{*}(B\mathbb V_s, F_2))\to {\rm Ext}_{\mathscr
{A}}^{s,s+*}(F_2, F_2),$ where $\mathbb V_s$ denotes the elementary abelian
$2$-group of rank $s$, and $H_*(B\mathbb V_s)$ is the homology group of the
classifying space of $\mathbb V_s,$ while $P_{\mathscr A}(H_{*}(B\mathbb V_s,
F_2))$ means the primitive part of $H_{*}(B\mathbb V_s, F_2)$ under the action
of $\mathscr A.$ It has been shown that $Tr_s$ is highly non-trivial and, more
precisely, that $Tr_s$ is an isomorphism for $s\leq 3.$ In addition, Singer
proved that $Tr_4$ is an isomorphism in some internal degrees. He was also
investigated the image of the fifth transfer by using invariant theory.
  In this note, we use another method to study the image of $Tr_5.$ More
precisely, by direct computations using a representation of $Tr_5$ over the
algebra $\Lambda,$ we show that $Tr_5$ detects the non-zero elements $h_0d_0\in
{\rm Ext}_{\mathscr A}^{5, 5+14}(F_2, F_2),\ h_2e_0\in {\rm Ext}_{\mathscr
A}^{5, 5+20}(F_2, F_2)$ and $h_1h_4c_0\in {\rm Ext}_{\mathscr A}^{5, 5+24}(F_2,
F_2).$ The same argument can be used for homological degrees $s\geq 6$ under
certain conditions.
