Math 91 Cheat Sheet
October 19, 2022
1
Definitions
Suppose you have a linear transformation f : Rn → Rm which is given by a matrix A. Then
the descriptions within each list are equivalent. Try to convince yourself that these are true,
either with a proof or with some simple examples!
Injective:
• The map f is 1 to 1
• In REF or RREF, the matrix has a pivot in every column
• Null A = 0
• Columns of A are linearly independent
• The matrix A has a left inverse
Example 1. The following matrix is injective but not surjective:


1 0
A = 0 1 .
0 0
Surjective:
• The map f is onto
• In REF or RREF, the matrix has a pivot in every row
• Im A = Rm
• Columns of A span Rm
• The matrix A has a right inverse
Example 2. The following matrix is surjective but not injective:
1 0 0
A=
.
0 1 0
Bijective:
• The map f is injective and surjective
• The RREF of A is I.
• The matrix A is invertible
• The matrix A has both a left and right inverse
• The determinant of the matrix is non-zero
Example 3. Out of the following matrices, A is bijective, while B is not.
1 0
1 −1
A=
,
B=
.
1 1
1 −1
Orthogonal matrices:
• Multiplication by A preserves the dot product, i.e. (Av) · (Aw) = v · w for all v and w
• A = v1 . . . vn where the columns v1 , ..., vn are orthogonal. That is, they are
pairwise perpendicular: vi · vi = 1 and vi · vj = 0 for i ̸= j.
• The map f preserves angles between vectors and lengths of vectors
Example 4. The following matrices are orthogonal. The matrix A does nothing, the matrix
B rotates by the angle α, and the matrix C reflects over the line x = y.
1 0
cos α − sin α
0 1
A=
,
B=
,
C=
.
0 1
sin α cos α
1 0
Exercise 5. Suppose I have an m × n matrix A which is injective. What does this tell me
about m and n? (Do the same thing for surjective and bijective.)
2
Useful principles for problem-solving
n
m
n
1. Any linear transformation
R → R isdetermined by what it does to a basis of R . More
concretely, we have: A = Ae1 . . . Aen
Example 6. Suppose you want to find the linear transformation given by a matrix which
rotates by 270 degrees. This sends e1 to −e2 and e2 to e1 . Thus the matrix is
0 1
A=
.
−1 0
2. Composing linear maps is the same as multiplying their matrices. More explicitly, if
two linear maps f, g are given by matrices Af , Ag , then Ag Af is the matrix of the composition
g ◦ f.
Example 7. Suppose you want to find all matrices such that A3 = I. This is the same as
finding all linear transformations f where applying f three times (f ◦ f ◦ f ) is the same as
the identity. These are exactly rotation by 0, 2π/3, and 4π/3.