Chapter 4 Review

Theorem: Let S = {a₁, …, a_m} be a set of vectors of ℝⁿ. Let A = [a₁ … a_n] be a matrix and T : ℝ^m → ℝⁿ be the linear transform defined by T(x) = Ax. Let B be an echelon form of A. Then the following objects are equal:

• The set of vectors killed by T,
• {x : Ax = 0} (this is the set of homogeneous solutions to A),
• null(A),
• {x : T(x) = 0},
• ker(T),
• number of rows of all zeros in B,

• The set of vectors hit by T,
• {T(x) : x ∈ ℝⁿ},
• range(T),
• col(A),
• span(S),

• dim(col(A)),
• dim(range(T)),
• dim(span(S)),
• m - nullity(A) (rank-nullity theorem),
• m - dim(ker(T)),
• dim(row(A)), (think of this as maximal number of linear independent equations in Ax = 0),
• number of pivots in B,

Example: Let T(x) = Ax, where A is

$\begin{bmatrix} 1 & 2 & 0 & 2 \\ -2 & -4 & 1 & -3 \\ 1 & 2 & 2 & 4 \end{bmatrix}$

and has reduce echelon form B given by

$\begin{bmatrix} 1 & 2 & 0 & 2 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}$

• What is the range of T?
• What is the kernel of T?
• What is the row space of A?
• What is the rank of A?
• What is the nullity of A?
• Write the columns corresponding the free variables as a linear combination of the pivot columns.
• What is the general solution to Ax = 0?
• What is the general solution to Ax = [2 − 3, 4]^t?
• What is a vector not in the range of T?

Example: Answer all the same questions as above but for an invertible transform.

Example: Give an example of a linear transform T : ℝ³ → ℝ² such that T(1, 1, 0) = (1, 0) and T(0, 1, 2) = (1, 2).

• What is the smallest possible rank such an example could be?
• What is the largest possible rank such an example could be?
• What is the smallest possible nullity such an example could be?
• What is the largest possible nullity such an example could be?