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Module 5: Quick Reference

Linear Transformations -- Linear Algebra

Definition

T : R^n → R^m is linear if T(u+v) = T(u)+T(v) and T(cu) = cT(u)
T(0) = 0 always. If T(0) is not 0, then T is NOT linear.

Standard Matrix

A = [T(e1) | T(e2) | ... | T(en)], so T(x) = Ax

Rotation by theta

[cos(theta) -sin(theta); sin(theta) cos(theta)]

Reflection over x-axis

[1 0; 0 -1]

Projection onto x-axis

[1 0; 0 0]

Scaling by k

[k 0; 0 k]

Kernel and Range

Kernel (Null Space)

ker(T) = {x : Ax = 0}

Nullity = dim(ker(T)) = # free variables

Range (Image)

range(T) = col(A)

Rank = dim(range(T)) = # pivot columns

Rank-Nullity Theorem

rank(T) + nullity(T) = n (number of columns)

One-to-One

ker(T) = {0}, nullity = 0

Onto

range(T) = R^m, rank = m