Prove that $V={(𝑥_1,𝑥_2,⋯,𝑥_n) \in ℝ^n ∣ 𝑥_1+𝑥_2+...+𝑥_{𝑛−1}−2𝑥_𝑛=0}\}$ is a subspace of $\R^n$.
Answer
Let $t,s \in \R$ and $x,y\in V$. Then $x=(x_1, \dots, x_{n-1}, x_n)$ and $y=(y_1, \dots, y_{n-1}, y_n)$ with
\[x_1 +x _2 +...+x_{n−1} −2x_ n =0\]
and
\[y_1 +y _2 +...+y_{n−1} −2y_ n =0.\]
Hence
\[z=tx+sy=t(x_1, \dots, x_{n-1}, x_n)+s(y_1, \dots, y_{n-1}, y_n)\]
\[=(tx_1+sy_1, tx_2+sy_2, \dots, tx_{n}+sy_{n}).\]
Thus we have
\[z_1 +z _2 +...+z_{n−1} −2z_ n=\]
\[(tx_1+sy_1)+(tx_2+sy_2)+\dots +(tx_{n-1}+sy_{n-1})-2 (tx_{n}+sy_{n})\]
\[=t(x_1 +x _2 +...+x_{n−1} −2x_ n)+s(y_1 +y _2 +...+y_{n−1} −2y_ n )=t\times 0+s\times 0=0.\]
Therefore $tx+sy \in V$ and hence $V$ is a vector space.

4.8K
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- answered
- 1734 views
- $15.00
Related Questions
- Find the values of x
- Decide if the following representations are linear representations.
- Find a vector parametric form and symmetric form, find minimal distance betwen L and P, consider vectors v and w.
- [Linear Algebra] Proof check. Nilpotent$\Rightarrow Spec\Rightarrow$ Characteristic Polynomial $\Rightarrow$ Nilpotent
- Calculate the inverse of a triangular matrix
- Certain isometry overfinite ring is product of isometries over each local factor
- Find where this discrete 3D spiral converges in explict terms
- Determine and compute the elementary matrices: Linear Algebra