Tutorial # 3, MATH 1104B/MATH1107B, Fall 2025
Oct. 8, 2025
1. Determine whether the following transformations are linear transformations or not. Justify your answers.
x+y
x
y
x
(a) T
=
(b) T
=
y
x−y
y
x2
2. Let T : R2 → R2 be the transformation that reflects each vector
(x1 , x2 , x3 ) through the plane x3 = 0 onto T (x) = (x1 , x2 , −x3 ). Show
that T is a linear transformation.
3. Let T : R2 → R2 be the linear transformation which rotates vectors
π/3 radians counterclockwise. Find the standard matrix of T .
4. Let T : R4 → R3 be a linear transformation defined by
x
x
−
y
+
z
+
2w
y
T
z = 2x + y + 5z + w .
x + 2y + 3z + w
w
a) Find the standard matrix of T . b) Is T one-to-one? c) Is T onto?
d) Find all possible vectors x in R4 such that T (x) = 0.
5. Truth or False questions.
(a) If T : Rn → Rm be a linear transformation with the standard matrix
A, then T is one to one if and only if A has n pivot columns.
(b) If T : Rn → Rm be a linear transformation with the standard
matrix A, then T is onto if and only if A has m pivot columns.
(c) Let T : Rn → Rm be a linear transformation. Then T is onto if and
only if the range of T is equal to the codomain of T .
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