Changes

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More concisely, we could write <math>x^i</math> for the components of the vector, where <math>i=1,2,3</math>. To motivate this notation, we will consider the equation <math>Ax=b</math> for some matrix <math>A</math> and vectors <math>x</math> and <math>b</math>. We will use the convention that if <math>A</math> is a matrix, then <math>(A)_{ij}=a_{ij}</math> is the element of that matrix in the ith row and jth column. One way of thinking of vector equations is as a shorthand for a set of simultaneous equations - each component of the vectors gives an equation. Explicitly, consider the set of three equations for the three unknowns <math>x^1, x^2, x^3</math>:

More concisely, we could write <math>x_i</math> for the components of the vector, where <math>i=1,2,3</math>. To motivate this notation, we will consider the equation <math>Ax=b</math> for some matrix <math>A</math> and vectors <math>\vec{\boldsymbol{x}}</math> and <math>\vec{\boldsymbol{b}}</math>. We will use the convention that if <math>A</math> is a matrix, then <math>(A)_{ij}=a_{ij}</math> is the element of that matrix in the ith row and jth column. One way of thinking of vector equations is as a shorthand for a set of simultaneous equations - each component of the vectors gives an equation. Explicitly, consider the set of three equations for the three unknowns <math>x_1, x_2, x_3</math>:

:<math>

:<math>

a_{11} x^1+a_{12} x^2+a_{13} x^3 = b^1

a_{11} x_1+a_{12} x_2+a_{13} x_3 = b_1

</math>

</math>

:<math>

:<math>

a_{21} x^1+a_{22} x^2+a_{23} x^3 = b^2

a_{21}x_1+a_{22}x_2+a_{23}x_3 = b_2

</math>

</math>

:<math>

:<math>

a_{31} x^1+a_{32} x^2+a_{33} x^3 = b^3

a_{31}x_1+a_{32}x_2+a_{33}x_3 = b_3

</math>

</math>