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Exercise For u = (1, 2), v = (3, 4), w = (5, 6), compute the linear combination x1u + x2v + x3w. Do the same for a1 = (1, 2, 3), a2 = (4, 5, 6), a3 = (7, 8, 9). Express your computation in vertical vectors and draw a general observation.

Answer

x1[ 1 ] + x2[ 3 ] + x3[ 5 ] = [ x1 ] + [ 3x2  ] + [ 5x3 ] = [ x1 + 3x2 + 5x3 ]
2 4 6 2x1 4x2 6x3 2x1 + 4x2 + 6x3
x1[ 1 ] + x2[ 4 ] + x3[ 7 ] = [ x1 ] +[ 4x2 ] + [ 7x3 ] = [ x1 + 4x2 + 7x3 ]
2 5 8 2x1 5x2 8x3 2x1 + 5x2 + 8x3
3 6 9 3x1 6x2 9x3 3x1 + 6x2 + 9x3

The results are of the form Ax, with the matrix A being respectively

[ 1 3 5 ] = [u v w]
2 4 6
[ 1 4 7 ] = [a1 a2 a3]
2 5 8
3 6 9

In general, we have

x1a1 + x2a2 + ... + xnan = Ax

where A = [a1 a2 ... an] is a matrix with the vectors a1, a2, ..., an as columns.