Rather than use the standard definitions of addition and scalar multiplication in R3, suppose these two operations are defined as follows. With these new definitions, is R³ a vector space? Justify your answers.
⁽ˣ¹, ˣ¹, ᶻ¹⁾ ⁺ ⁽ˣ², ˣ², ᶻ²⁾ ⁼ ⁽ˣ¹ ⁺ ˣ², ʸ¹ ⁺ ʸ², ²¹ ⁺ ᶻ²⁾ ᶜ⁽ˣ, ʸ, ᶻ⁾ ⁼ ⁽ᶜˣ, ⁰, ᶜᶻ⁾
• The set is a vector space.
• The set is not a vector space because the associative property of addition is not satisfied.
• The set is not a vector space because it is not closed under scalar multiplication.
• The set is not a vector space because the associative property of multiplication is not satisfied.
• The set is not a vector space because the multiplicative identity property is not satisfied.