Given a vector x and a matrix A, we want to define their product Ax. (This operation can be used to represent things like rotations in video games.)
A=⎝⎛341254163⎠⎞, x=⎝⎛123⎠⎞Ax=⎝⎛341254163⎠⎞⎝⎛123⎠⎞
Rotate each row of the matrix so it becomes a column, then multiply that column element-wise with the vector and add the results.
Ax=⎝⎛3(1)+2(2)+1(3)4(1)+5(2)+6(3)1(1)+4(2)+3(3)⎠⎞=⎝⎛3+4+34+10+181+8+9⎠⎞=⎝⎛103218⎠⎞
The number of columns in the matrix must be the same as the number of rows in the vector for this operation to be defined.
If A is a n by m matrix with columns a1...an and x is a column vector with length m, then
Ax=x1a1+...+xnan=b,
whose solution set is the same as the solution set of the equations whose augmented matrix is [a1...an,b].
Ex. Let A=⎝⎛1−4−332−24−6−7⎠⎞.
Can we choose b in R3 in an arbitrary manner and still have Ax=b be consistent? (This is logically equivalent to the question, "Do the columns of A span R3?")
b=Augmented matrix: ==⎝⎛b1b2b3⎠⎞⎝⎛1−4−332−24−6−7b1b2b3⎠⎞⎝⎛10031474105b1b2+4b1b3+3b1⎠⎞⎝⎛10031404100b1b2+4b1b3+3b1−21(b2+4b1)⎠⎞
Based on the bottom row, we can see that for some choices of b, this system will be inconsistent; therefore the columns of A do not span R3.