# Philosophy, Computing, and Artificial Intelligence

PHI 319. Vectors and Matrices.

## Vectors and Row Vectors

For an introduction to vectors and linear algebra, see Essence of Linear Algebra, 3Blue1Brown. A m-vector is a list of length m

[[x_1],[x_2],[.],[.],[.],[x_m]]

A m-row vector is the transpose of a m-vector

[[x_1],[x_2],[.],[.],[.],[x_m]]^T

## The Transpose Operator

The transpose operator T transposes a m-vector into a m-row vector (and vice versa)

[[x_1],[x_2],[.],[.],[.],[x_m]]^T = [x_1,x_2,.,.,.,x_m]

[x_1,x_2,.,.,.,x_m]^T = [[x_1],[x_2],[.],[.],[.],[x_m]]

## The Dot Product

If bbx and bby are m-vectors, the dot product bbx * bby is a number

bbx^T bby = [x_1,x_2,.,.,.,x_m] [[y_1],[y_2],[.],[.],[.],[y_m]] = x_1 y_1 + x_2 y_2 + ... + x_m y_m = sum_i^m x_i y_i

If bbW is a m x n matrix and bbx is a n-vector, the dot product is a m-vector

 bbW * bbx = [ [w_(11), w_(12), w_(13), ., ., ., w_(1n)], [w_(21), w_(22), w_(23), ., ., ., w_(2n)],[.,.,.,,,,.],[.,.,.,,,,.],[.,.,.,,,,.], [w_(m1), w_(m2), w_(m3), ., ., ., w_(mn)] ] [ [x_1], [x_2], [.], [.], [.], [x_n]] = [ [w_(11)x_1 + w_(12)x_2 + ... + w_(1n)x_n], [w_(21)x_1 + w_(22)x_2 + ... + w_(2n)x_n], [.], [.], [.], [w_(m1)x_1 + w_(m2)x_2 + ... + w_(mn)x_n]]

If bbW is a m x n matrix and bbY is a n x 2 matrix, the dot product is a m x 2 matrix

 bbW * bbY = [ [w_(11), w_(12), w_(13), ., ., ., w_(1n)], [w_(21), w_(22), w_(23), ., ., ., w_(2n)],[.,.,.,,,,.],[.,.,.,,,,.],[.,.,.,,,,.], [w_(m1), w_(m2), w_(m3), ., ., ., w_(mn)] ] [ [x_(11), x_(12)], [x_(21), x_(22)], [.,.], [.,.], [.,.], [x_(n1), x_(n2)] ]

 = [ [w_(11)x_(11) + w_(12)x_(21) + ... + w_(1n)x_(n1), w_(11)x_(12) + w_(12)x_(22) + ... + w_(1n)x_(n2)], [w_(21)x_(12) + w_(22)x_(22) + ... + w_(2n)x_(n2), w_(21)x_(12) + w_(22)x_(22) + ... + w_(2n)x_(n2)], [.,.], [.,.], [.,.], [w_(m1)x_(1n) + w_(m2)x_(2n) + ... + w_(mn)x_(n n), w_(m1)x_(12) + w_(m2)x_(22) + ... + w_(mn)x_(n n)]]

An example makes this a little clearer.

 bbA = [[1,2,3],[4,5,6]]     bbB = [[1,2],[1,2],[1,2]]        bbA * bbB = [ [6, 15], [12, 30] ]