Standard Basis

Standard Basis Vectors

We refer to the vectors e_1=(1,0) and e_2=(0,1) as the standard basis vectors in R^2. A vector a=(a_1,a_2) can be expressed in term of the basis vectors e_1 and e_2 as

  • a=a_1e_1+a_2e_2,

since

  • a_1e_1+a_2e_2=a_1(1,0)+a_2(0,1)=(a_1,0)+(0,a_2)=(a_1,a_2)=a.

as illustrated in:

The standard basis vectors e_1 and e_2 and a linear combination a=(a_1,a_2)=a_1e_1+a_2e_2 of e_1 and e_2.

We say that a_1e_1+a_2e_2 is a linear combination of e_1 and e_2 with coefficients a_1 and a_2. Any vector a=(a_1,a_2) in R^2 can thus be expressed as a linear combination of the basis vectors e_1 and e_2 with the coordinates a_1 and a_2 as coefficients.

Example: We have (3,7)=3, (1,0)+7, (0,1)=3e_1+7e_2.

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