Vectors with the same number of components, n, can be added or subtracted element-wise to obtain another vector, where each element will be of the same size. These operations are performed element by element. Additionally, any vector can be multiplied by a scalar (real or complex number). This operation is performed by multiplying each vector element by the scalar, resulting in another vector of the same size.

4.3.1. Addition and Multiplication with Scalars
Let vectors u and v have components u₁, u₂, …, uₙ and v₁, v₂, …, vₙ, respectively. In this case, the sum u + v forms a vector with components uᵢ + vᵢ. If k is a scalar (a real or complex number), the vector multiplied by k will have components k * uᵢ.

For example, if the vectors u and v are given in ℂ² as follows:
u = (1 + i, 3 – 2i)
v = (7 – 2i, 8 – 3i)
Then:
u + v = (8 – i, 11 – 5i)

This property states that the sum of two vectors consisting of real or complex numbers, or a vector multiplied by a scalar, results in another vector of the same size and nature. This shows that vector addition and scalar multiplication operations are closed under these operations. In other words, the resulting vectors belong to the same type as the vectors performing the operation.

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Any vector in ℝ² can be constructed using the standard orthonormal basis vectors i and j. Similarly, vectors in ℝ³ can be constructed using the basis vectors i, j, and k.

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