Properties of linear spaces Consider the set of ordered pairs defined on the Cartesian product V...

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Properties of linear spaces Consider the set of ordered pairs defined on the Cartesian product V = {0, 1, 2} x {0, 1, 2}, i.e. pairs x = (x1, x2) EV with ₁ € {0, 1, 2} and x2 = {0, 1,2}. We can also refer to the pairs as 2-dimensional vectors. We define scalar multiplication of elements in V such that for a scalar a = {0, 1, 2}, we have a> x = (a x₁, a 2). Furthermore, vector addition between elements in V is defined such that xy = (21 y₁, 22 y2). The addition (0) and multiplication () operations for scalars are defined according to the following tables. 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1 0 1 2 00 1 2 0 0 1 2 0 0 21 i. Determine whether or not V, along with and defines a vector space (i.e., verify the 10 properties). ii. Find a subspace of V with dimension equal to 1, and verify that it is also a vector space. Properties of linear spaces Consider the set of ordered pairs defined on the Cartesian product V = {0, 1, 2} x {0, 1, 2}, i.e. pairs x = (x1, x2) EV with ₁ € {0, 1, 2} and x2 = {0, 1,2}. We can also refer to the pairs as 2-dimensional vectors. We define scalar multiplication of elements in V such that for a scalar a = {0, 1, 2}, we have a> x = (a x₁, a 2). Furthermore, vector addition between elements in V is defined such that xy = (21 y₁, 22 y2). The addition (0) and multiplication () operations for scalars are defined according to the following tables. 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1 0 1 2 00 1 2 0 0 1 2 0 0 21 i. Determine whether or not V, along with and defines a vector space (i.e., verify the 10 properties). ii. Find a subspace of V with dimension equal to 1, and verify that it is also a vector space.