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Unformatted text preview: University of Toronto at Scarborough
Department of Computer & Mathematical Sciences
MAT A23 Winter 2008
Lecture 16
Section 2.3
Sophie Chrysostomou 1 Linear Transformations theorem 1.1 If T : Rn −→ Rm is a linear transformation and v1 , v2 , · · · , vk ∈
Rn such that {T (v1 ), T (v2 ), · · · , T (vk )} is a linearly independent set in Rm .
Then, {v1 , v2 , · · · , vk } is linearly independent also.
proof. 1 theorem 1.2 (from Lecture 15) If T : Rn −→ Rm is a linear transformation, and B = {b1 , b2 , · · · , bn } a basis for Rn .
Then:
if v ∈ Rn =⇒ T (v) is determined by T (b1 ), T (b2 ), · · · , T (bn ).
OR T [Rn ] ⊂ sp(T (b1 ), T (b2 ), · · · , T (bn ).)
corollary 1.3
m × n matrix If T : Rn −→ Rm is a linear transformation, let A be the 


A = T (e1 ) T (e2 ) · · · T (en ) 

 Then T (x) = Ax for all x ∈ Rn . A is called the standard matrix representation of T .
(This is very powerful!!! It says that every linear transformation corresponds
to a matrix multiplication.) 2 example 1.4 Suppose T : R4 −→ R3 is a linear transformation given by
T ([x1 , x2 , x3 , x4 ]) = [x2 − 3x3 − x4 , 6x1 + 5x2 , x3 + 2x1 ]. Find the standard
matrix representation of T . Homework: Suppose T : R3 −→ R5 is a linear transformation given by
T ([x1 , x2 , x3 ]) = [2x1 + 3x3 − 4x2 , 3x1 − 2x3 , 7x2 + 4x3 , x1 − 4x2 + 2x3 , 5x2 ].
Find the standard matrix representation of T .
Question: Let T : Rn −→ Rm be a linear transformation. How do we ﬁnd
the kernel of T ? How do we ﬁnd the range of T ?
Answer: Let A be the standard matrix representation of T . So T (x) = Ax
for all x ∈ Rn .
Remember, by deﬁnition: The nullspace or kernel of T is given by
ker(T ) = T −1 [{0 }] = {v ∈ Rn T (v) = 0}
= {v ∈ Rn Av = 0}
= nullspace of A
The range of T is given by
T [Rn ] = {T (x) x ∈ Rn }
= { Ax x ∈ R n }
= column space of A
3 So:
i) to ﬁnd the nullspace (kernel) 0f T we ﬁnd the nullspace of A.
ii) to ﬁnd the range of T , we ﬁnd the column space of A.
example 1.5 Suppose T : R4 −→ R3 is a linear transformation given by
T ([x1 , x2 , x3 , x4 ]) = [x2 − 3x3 − x4 , 6x1 + 5x2 , x3 + 2x1 ]. Find a basis for the
nullspace and a basis for the range of T . 4 definition 1.6 Let T : Rn −→ Rm be a linear transformation.
• rank of T , denoted by rank (T ), is the dimension of the range of T .
• nullity of T , denoted by nullity (T ), is the dimension of the kernel of
T.
• T is invertible if m = n and the standard matrix representation A of T
is invertible.
• the rank equation for linear transformations is:
rank (T ) + nullity (T ) = dim(domain of T )
example 1.7 Let T : R4 −→ R4 be the linear transformation given by
T ([x1 , x2 , x3 , x4 ]) = [x1 + x2 , x2 + x3 , −x4 + 2x2 + x3 , x1 + x4 ]. Find a basis
for the range of T , a basis fo the nullspace of T , verify the rank equation,
and ﬁnd if T is invertible. 5 ...
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 Spring '10
 Sophie
 Linear Algebra, Algebra, Transformations

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