Write every vector as sum of vectors from these subspaces

Question

$$S = \{(x_1, x_2, ...,x_n)^T~|~ x_1+2x_2+3x_3+...+nx_n=0 \}$$

$$T = \{(x_1, x_2, ...,x_n)^T~|~ x_1=x_2=x_3=...=x_n \}$$

$R^n$ is a direct sum of $S$ and $T$. This follows from Grassmann formula:

$\dim S + \dim T = \dim(S\cap T) + \dim(S + T)$

In other words every vector $v\in$ $R^n$ can be written as $v=s+t$, where $s\in S, t \in T$.

Writing out the canonical basis is sufficient. For example :

$$(1,0,0,...,0)=t+s$$

Can someone help me find $s$ and $t$ .

Answer 1

If $n=4$, then:

• $\displaystyle(1,0,0,0)=\overbrace{\left(\frac9{10},-\frac1{10},-\frac1{10},-\frac1{10}\right)}^{\phantom S\in S}+\overbrace{\left(\frac1{10},\frac1{10},\frac1{10},\frac1{10}\right)}^{\phantom T\in T}$;
• $\displaystyle(0,1,0,0)=\overbrace{\left(-\frac2{10},\frac8{10},-\frac2{10},-\frac2{10}\right)}^{\phantom S\in S}+\overbrace{\left(\frac2{10},\frac2{10},\frac2{10},\frac2{10}\right)}^{\phantom T\in T}$
• $\displaystyle(0,0,1,0)=\overbrace{\left(-\frac3{10},-\frac3{10},\frac7{10},-\frac3{10}\right)}^{\phantom S\in S}+\overbrace{\left(\frac3{10},\frac3{10},\frac3{10},\frac3{10}\right)}^{\phantom T\in T}$
• $\displaystyle(0,0,0,1)=\overbrace{\left(-\frac4{10},-\frac4{10},-\frac4{10},\frac6{10}\right)}^{\phantom S\in S}+\overbrace{\left(\frac4{10},\frac4{10},\frac4{10},\frac4{10}\right)}^{\phantom T\in T}$.

Can you deal with the general case now?