Mathematics: Let v be the vector space of all 2×2 matrices with real entries. let h be the set of all 2×2 idempotent matrices with real entries. is h a subspace of the vector space v? does h contain the zero vector of v?

Let v be the vector space of all 2×2 matrices with

GalleTF

09.01.2022

Remember, a set $S\subset V$ is a subspace of the vector space V if:

$1. 0\in S\\2. a+b\in S,\text{ for all } a,b\in S\\3. \lambda a\in S, \text{for } \lambda \text{ a scalar}$ .

1. Observe that

$0\times 0=0^2=0$ , where is the null matrix $2\times 2$ .

Then $0 \in H$

2. Consider two matrices A and B such that $AB\neq 0$ in H

Observe that $(A+B)^2=A^2+2AB+B^2\\=A+2AB+B\neq A+B$ .

Then, $A+B\nsubseteq H$

Then H is not a subspace of V

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graciecope04

09.04.2023

1/5 a foot taller

Step-by-step explanation: Basically you just subtract 2/5 from 3/5. That gives you 1/5.

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