The document defines kernel (ker(T)) as the set of vectors mapped to 0 by a linear transformation T. The range (R(T)) is defined as the set of vectors in the codomain that are images of vectors in the domain. Examples are given to illustrate kernel and range for different types of linear transformations such as the zero transformation, orthogonal projection onto a plane, and rotation. The rank of a matrix is defined as the dimension of its row space, while the nullity is defined as the dimension of its null space. Steps are shown to compute the rank and nullity of matrices by putting them in reduced row echelon form.

2. ker(T): the kernel of T
If T:V→W is a linear transformation, then the set of vectors in V that T maps into 0.
R (T): the range of T
The set of all vectors in W that are images under T of at least one vector in V.

3. If TA : 𝑅𝑛
→ 𝑅𝑚
is a multiplication by the m×n matrix A, from the discussion preceding
the definition above,
is the nullspace of A
Is the column space of A

4. Let T : V → W be the zero transformation.
Since T maps every vector in V into 0, it follows that ker (T) = V
Moreover, since 0 is the only image under T of vectors in V, we
have R (T) = 0

5. Let T : 𝑅3
→ 𝑅3
be the orthogonal projection on the xy-plane.
The kernel of T is the set of points that T maps into 0 = (0,0,0);
these are the points on the z-axis.

6. Since T maps every points in 𝑅3 into the xy-plane, the range of T
must be some subset of this plane. But every point (X0 , Y0 , 0) in the xy-
plane is the image under T of some point; in fact, it is the image o all points
on the vertical line that passes through (X0 , Y0 , 0) . Thus, R (T) is the entire
xy-plane.

7. Let T : 𝑅2
→ 𝑅2
be the linear operator that rotates each vector in the xy-
plane through the angle 𝜃. Since every vector in the xy-plane can be
obtained by rotating through some vector through angle 𝜃, we have R (T) =
𝑅2
. Moreover, the only vector that rotates into 0 is 0, so ker(T) = 0

8. If T : V → W is linear transformation, then:
Of T is a subspace of V
The
The Of T is a subspace of W

9. If A is any matrix, then the row space and column space of A
have the same dimension

10. The of the matrix A is the dimension of the row space
of A, and is denoted by R(A).
The of a matrix A is the dimension of the null space
of A, and is denoted by N(A). Let A be an m x n matrix. The
null space is the set of solutions to the homogenous system
Ax=0.

11. A =
1 2 4
2 4 8
Step 1. Transform the matrix into REF
A =
1 2 4
2 4 8
R → R2 – 2R1
A =
1 2 4
0 0 0
2 – 2(1) = 0
4 – 2(2) = 0
8 – 2(4) = 0
Rank = r(A) = 1 because only 1 row has non-zero elements
nullity = n(A)
n(A) = n – r(A)
n(A) = 3 – 1
n(A) = 2
r(A) = 1
n(A) = 2

12. A =
1 4 5
0 1 3
0 0 0
r(A) = 2 (because 2 rows have
non-zero elements)
n(A) = n – r(A)
= 3 – 2
= 1
r(A) = 2
n(A) = 1

13. Find the Rank and Nullity
1.) A =
1 4 6
2 8 12
2.) B =
1 4 6
0 1 7
0 0 0
Solutions
1.) A =
1 4 6
2 8 12
R2→2R1
A =
1 4 6
0 0 0
2 – 2(1) = 0
8 – 2(4) = 0
12 -2(6) = 0
r(A) = 1
n(A) = 2
2.) B =
1 4 6
0 1 7
0 0 0
r(B) = 2
n(B) = 1