1. MATH 1046 Introduction to Linear Transformations (Section 3.6)
Alex Karassev

2. Terminology
The following mean essentially the same and can be used interchangeably:
Linear map
Linear function
Linear transformation

3. Definition Let U and V be vector spaces
A function T: U  V is called a linear transformation if for all x,y from U and all real numbers a we have
T(x+y) = T(x) + T(y)
T(ax) = aT(x)

4. Equivalent Definition
Let U and V be vector spaces
A function T: U V is called a linear transformation if for all u1, u2,…, uk and all real numbers c1, c2, … ck we have T(c1u1+c2u2+…+ckuk)=c1T(u1)+c2T(u2)+…+ckT(uk)

5. Non-example Let U=V=R Let T(x) = x2
Then T is not a linear transformation: e.g. T(1+2) = 32 ≠ 12+22=T(1)+T(2)

6. Example
Any linear transformation T: RR has the form T(x)=mx for some real number m (exercise: prove it)
Thus most functions on R are not linear transformations

7. Why then do we study linear transformations?
Any differentiable function from R to R can be approximated by a function of the form mx+b (linear transformation + constant) at every point of its graph; similarly for functions on Rn
Many geometric transformations of the plane or space (such as reflections, rotations, projections, scaling etc.) are linear transformations; they have applications in particular in computer graphics
Many operations in mathematics (e.g. differentiation or integration) are linear and can be viewed as linear transformations of appropriately chosen vector spaces

8. Range and Kernel Let T : U V be a linear transformation
Range of T is the set T(U) = {T(u) | u in U}
Kernel of T is the set Ker (T) = {u in U | T(u)=0}
Note: range is a subset of V and kernel is a subset of U

9. Theorem
Range and kernel of a linear transformation are subspaces of the corresponding vector spaces

10. Proof: range is a subspace of V
T : UV
T(U) = {T(u) | u in U}
Since T(0U) = T(0∙0U) = 0∙T(0U) = 0V, 0V is in T(U)
Note: as a by-product, we get T(0) = 0 for any linear transformation
If u and v are in T(U), there are x and y from U such that u=T(x) and v= T(y), so u+v = T(x)+ T(y) = T(x+y) is in T(U) cu = cT(x) = T(cx) is in T(U)

11. Proof: kernel is a subspace of U
T : UV
Ker(T) = {u in U | T(u) = 0}
Since T(0) = 0, 0 is in Ker(T)
Suppose u and v are in Ker (T)
Then T(u) = 0 and T(v) = 0
Therefore T(u+v) = T(u) + T(v) = = 0 so u+v is in Ker(T) and T(cu) = c T(u) = c0 = 0 so cu is in Ker(T)

12. Linear transformations and matrices
Let T : R2  R2
𝒆 𝟏 = , 𝒆 𝟐 = 0 1
T( 𝒆 𝟏 )= 𝑎 11 𝑎 21 , T( 𝒆 𝟐 )= 𝑎 12 𝑎 22
Then for any 𝒙= 𝑥 1 𝑥 2 we have T(𝒙)=T( 𝑥 1 𝒆 𝟏 + 𝑥 2 𝒆 𝟐 )= 𝑥 1 𝑇( 𝒆 𝟏 )+ 𝑥 2 𝑇( 𝒆 𝟐 )= 𝑥 1 𝑎 11 𝑎 𝑥 2 𝑎 12 𝑎 22 = 𝑎 11 𝑎 12 𝑎 21 𝑎 22 ∙ 𝑥 1 𝑥 2 =𝐴𝒙
Thus T(x) = Ax

13. In general A matrix of a given linear transformation
Let T: Rn Rm be a linear transformation
Let AT be m x n matrix whose columns are the coordinates of the images of the standard basis: AT= [T(e1), T(e2),…, T(en)]
Then for any x in Rn we have T(x) = ATx
Linear transformation given by a matrix
Let A be any m x n matrix
Then TA (x) = Ax is a lin. transformations from Rn to Rm
Indeed: TA(x+y) = A(x+y) = Ax + Ay= TA(x) + TA(y) and TA(cx) = c TA(x)

14. Example: find a matrix of a counter-clockwise rotation by 45 degrees

15. Example: find a matrix of projection
Find matrices of projections onto the following lines
the x2-axis
the line x1 = x2
Determine the range and kernel for each of these transformations

16. Exercises
A linear transformation of R2 to R2 maps lines to lines or points
A linear transformation of R3 to R3 maps lines to lines or points and planes to planes, lines or points
Any linear transformation T: U V maps subspaces of U to subspaces of V
Does a linear transformation T maps any basis of U to a basis of V?

17. Kernel, Range, and matrices
Let T(x) = Ax
Ker (T) = { x in Rn | T(x) = 0} = { x in Rn | Ax =0} = nullspace (A)
T(Rn) = {T(x) | x in Rn} = { Ax | x in Rn} = col(A)

18. Example
Consider a (non-linear) map H: R2 R2 given by H(x,y) = (f(x,y),g(x,y))=(x2- y2, 2xy)

19. H(x,y) = (x2- y2, 2xy)
Where does H map the lines parallel to the coordinate axis?

20. H(x,y) = (x2- y2, 2xy)
Exercise: find the equations of the images of the vertical and horizontal lines

21. H(x,y) = (x2- y2, 2xy) What happens near (1,1)? Note: H(1,1) = (0,2)
Zoom

22. H(x,y) = (x2- y2, 2xy)
It appears that on a small scale near (1,1) the map H can be approximated by a counter-clockwise rotation by 45 degrees. Is it really the case?

23. Matrix of partial derivatives
H(x,y) = (f(x,y),g(x,y))=(x2- y2, 2xy)
𝜕𝑓 𝜕𝑥 𝜕𝑓 𝜕𝑦 𝜕𝑔 𝜕𝑥 𝜕𝑔 𝜕𝑦 = 2𝑥 −2𝑦 2𝑦 2𝑥
At x=1, y=1, we get 2 − = 23/2 1/ 2 −1/ 2 1/ 2 1/ which is a counter-clockwise rotation by 45 degrees followed by a scaling by a factor of 23/2