Review• A linear map T : V →W satisfies T (cx + dy) = cT (x) + dT (y).• T : Rn→Rmdefined by T (x) = Ax is linear. (A an m × n m atrix)◦ A is the matrix representing T w.r.t. the standard basesFor instance:T (e1) = Ae1= 1stcolumn of A e1=100• Let x1,, xnbe a basis for V , and y1,, yma basis for W .◦ The matrix rep resenting T w.r.t. these bases encodes in column j the coefficients of T (xj)expressed as a linear combination of y1,, ym.◦ For instance: let T : R3→ R3be reflecti on through the x-y-plane , that i s, (x, y, z)(x, y,−z).The matrix representingT w.r.t. the basis111,010,001is1 0 00 1 0−2 0 −1.T 111!=11−1=111+ 0010− 2001T 010!=010, T 001!=00−1Example 1. Let T : P3→P2be the linear map given byT (p(t)) =ddtp(t).What is the ma t r ix A representing T with respect to the standard bases?Solution. The bases are1, t, t2, t3for P3, 1, t, t2for P2.The matrix A h as 4 columns and 3 rows.Armin [email protected] first column encodes T (1) = 0 and hence is000.For the second column,T (t) = 1 and hence it i s100.For the third column,T (t2) = 2t and hence it is020.For the last column,T (t3) = 3t2and henc e it is003.In conclusion, t he matrix representingT isA =0 1 0 00 0 2 00 0 0 3.Note: By the way, what is the null space o f A?The null space has basis1000. The corresponding polynomial is p(t) = 1.No surprise here: differentati on kills precisely the constant pol ynomials.Note: Let us differentiate7t3−t + 3 using the matrix A.• First: 7t3− t + 3 w.r.t. standard basis:3−107.•0 1 0 00 0 2 00 0 0 33−107=−102 1•−102 1in the standard basis is −1 + 21t2.Armin [email protected] inner product and distancesDefinition 2. The inner product (or dot product) of v, w in Rn:v ·w = vTw = v1w1++ vnwn.Be cause we can t hink of this as a special case of the matrix product, it satisfies the basic rules likeassociativity and distri butivity.In addition:v · w = w · v.Example 3. For instance,123·1−1−2= [1 2 3]1−1−2= 1 −2 −6 = −7.Definition 4.• The norm (or length) of a vector v in Rniskv k = v ·v√= v12++ vn2p.This is the distance to the origin.• The distance between points v and w in Rnisdist(v , w) = kv −w k.vwv − wExample 5. For instance, in R2,distx1y1,x2y2=x1− x2y1− y2= (x1− x2)2+ (y1− y2)2p.Armin [email protected] ogonal vectorsDefinition 6. v and w in Rnare orthogonal ifv ·w = 0.How is this related to our understanding of right angle s?Pythagoras:v and w are orthogonalkv k2+ kwk2= kv −w k2v ·v + w ·w = (v −w) ·(v −w)v · v − 2v· w +w· wv ·w = 0vwv − wExample 7. Are the foll owing vectors orthogonal?(a)12,−2112·−21= 1 ·(−2) + 2 ·1 = 0. So, yes, they are orthogonal.(b)121,−211121·−211= 1 ·(−2) + 2 ·1 + 1 ·1 = 1. So not orthogonal.Armin
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