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>\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>:

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>\boldsymbol{A}\,\vec{\boldsymbol{x}}=\vec{\boldsymbol{b}}</math> for some matrix <math>\boldsymbol{A}</math> and vectors <math>\vec{\boldsymbol{x}}</math> and <math>\vec{\boldsymbol{b}}</math>. We will use the convention that if <math>\boldsymbol{A}</math> is a matrix, then <math>(\boldsymbol{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>

This can be rewritten in a matrix/vector form as equation <math>Ax=b</math>:

This can be rewritten in a matrix/vector form as equation <math>\boldsymbol{A}\,\vec{\boldsymbol{x}}=\vec{\boldsymbol{b}}</math>:

:<math>

:<math>