P + Q = P + R where P, Q and R are subspaces. Should Q = R?

Question

Suppose that P, Q and R are subspaces of a vector space V such that P + Q = P + R. Should Q = R necessarily?

I have an intuition that the answer is no, but I couldn't find proper examples to illustrate the same. Any help would be very appreciated.

Answer 1

Since it is not stated that $P$ and $Q$ resp. $P$ and $R$ intersect trivially, here is another class of counterexamples:

Let $P = V$. Then for any choice of $Q$ and $R$, one has $P+Q = P+R = V$. Hence it suffices to choose different $Q$ and $R$.

Answer 2

The answer is no. Take in $\mathbb{R}^2$:

$P = \{(x,0),x\in \mathbb{R}\}$ ($x$-axis) and

$Q= \{(0,y);y\in \mathbb{R}\}$ ($y$-axis) and

$R = \{(x,x);x\in \mathbb{R}\}$.

Answer 3

No. Let $V = \mathbb{R}^2$, and let $P$ be the y-axis, $Q$ be the x-axis, and $R$ be the $y=x$ line. Then $P+Q = P+R = \mathbb{R}^2$, but $Q \neq R$.

Answer 4

Counterexample:

Let $e_1$, $e_2$ - basis in $\mathbb{R}^2$.

$P = \{ a e_1, a \in \mathbb{R}^2 \}$, $Q = \{ a e_2, a \in \mathbb{R}^2 \}$, $R = \{ a (e_1+e_2), a \in \mathbb{R}^2 \}$.

Answer 5

No. Take $P=V$, and then you will always have $P+R=Q+R$, no matter which subspaces $Q$ and $R$ are.