# Existence of bases $V$ and $W$ such that with respect to the bases, all entries of M(T)=1

I have reviewed a solution to the problem for the implication in one direction: Suppose $$V$$ and $$W$$ are ﬁnite-dimensional and $$T \in \mathcal{L}(V,W)$$. Prove that dimrange$$T=1$$ if and only if there exist a basis of $$V$$ and a basis of $$W$$ such that with respect to these bases,all entries of $$\mathcal{M}(T)$$ equal 1.

proving the backwards direction

Let $$v_1,...,v_m$$ be a basis for $$V$$

Let $$w_1,...,w_n$$ be a basis for $$W$$

Assume all entries of $$\mathcal{M}(T)$$ are 1 with respect to these bases

Define a linear map $$T:V\rightarrow W$$ by

$$T(v_i)=w_1+w_2+\ldots +w_n$$ for each $$i \in \{1,2,...,m\}$$

Hence $$\text{RangeT}=\text{Span}(w_1+w_2+ ... +w_n)$$

and dimRange$$T=1$$

I understand the solution in the part where all entries are equal to $$1$$ since the coefficients for the linear combinations of the basis vectors of $$W$$ will all be $$1$$.

However I cannot understand why this implies Range$$T=\text{Span}(w_1+\ldots+w_n)$$ since the span of $$w_1+\dots +w_n$$ should consist of all scalar multiples of the vector. Can someone explain the reason for this?

• Why in this problem is RangeT$=\text{Span}(w_1+\dots+w_m)$ I am really confused Since every vector in the range of the linear transformation is $w_1+\dots+w_m$ there are no scalar multiples of this vector?
– user736276
Jan 1, 2020 at 19:31
• Note that in the proof, we have only plugged in $v = v_i$ for $i = 1,\dots,m$. Taking linear combinations of these vectors will produce multiples of this vector. Jan 1, 2020 at 19:33

Any vector in $$V$$ can be written in the form $$v = a_1 v_1 + \cdots a_n v_n$$. With that, we see that \begin{align} T(v) &= T(a_1 v_1 + \cdots + a_m v_m) = a_1 T(v_1) + \cdots + a_m T(v_m) \\ & = a_1(w_1 + \cdots + w_n) + \cdots + a_m(w_1 + \cdots + w_n) \\ & = (a_1 + \cdots + a_m)(w_1 + \cdots + w_n). \end{align} So, we can indeed say that for any $$v \in V$$, $$T(v)$$ is a multiple of $$(w_1+ \cdots + w_n)$$ and thus in the span of $$w_1 + \cdots + w_n$$.