QS-021 · Linear Algebra · Easy
NPTEL GATE DA MOCK | Week 1 - 59
Let $S_1$ and $S_2$ be two subspaces of $\mathbb{R}^{10}$ with dimensions 7 and 3, respectively. Let $d$ be the dimension of $S_1 \cap S_2$.
Which of the following statements is/are correct?
1. The maximum possible value of $d$ is 7.
2. The maximum possible value of $d$ is 3.
3. The minimum possible value of $d$ is 1.
4. The minimum possible value of $d$ is 0.
Let $S_1$ and $S_2$ be two subspaces of $\mathbb{R}^{10}$ with dimensions 7 and 3, respectively. Let $d$ be the dimension of $S_1 \cap S_2$.
Which of the following statements is/are correct?
1. The maximum possible value of $d$ is 7.
2. The maximum possible value of $d$ is 3.
3. The minimum possible value of $d$ is 1.
4. The minimum possible value of $d$ is 0.
Solution
Assuming $S_1 S_2$ means the intersection $S_1\cap S_2$:
For two subspaces,
$$ \dim(S_1\cap S_2) \geq \dim(S_1)+\dim(S_2)-\dim(\mathbb{R}^{10}) $$ $$ d\geq 7+3-10=0. $$
Also, $d\leq \min(7,3)=3. $
Therefore, 0≤d≤3
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