matrix of linear transformation topic

2. FAZAIA
COLLEGE OF
EDUCATION
FOR WOMEN
PRESENTED BY:BEENISH
EBAD
PRESENTED TO: MAM
MEHAK
COURSE: ADV MATH-II
SEMESTER: 06
CLASS:BS.ED(H)

4.  relating to, resembling, or having a graph
that is a line and especially a straight
line involving a single dimension
 E.g y=mx + b
Linear:

5. Transformation:
 The operation of changing (as by rotation
or mapping) one configuration or
expression into another in accordance
with a mathematical rule.

6. Linear transformation
A linear transformation is a function from
one vector space to another that respects
the underlying (linear) structure of each
vector space. A linear transformation is also
known as a linear operator or map.

7. Matrix of linear transformation:
 The matrix of a linear transformation is a
matrix for which T (𝒙 ) =A𝒙 , for a
vector x⃗ in the domain of T. This means
that applying the transformation T to a
vector is the same as multiplying by this
matrix. Such a matrix can be found for any
linear transformation T from 𝑹 𝒏 to𝑹 𝒎, for
fixed value of n and m, and is unique to the
transformation.

8. Example: 01
Find the matrix of a linear transformation
𝑻: 𝑹 𝟑 → 𝑹 𝟒 defined by 𝑻 𝒙 𝟏, 𝒙 𝟐, 𝒙 𝟑 = 𝒙 𝟏 +
Standard basis of 𝑹 𝟑 = 𝟏, 𝟎, 𝟎 , 𝟎, 𝟏, 𝟎 , 𝟎, 𝟎, 𝟏
= 𝒆 𝟏, 𝒆 𝟐, 𝒆 𝟑
Standard basis of 𝑹 𝟒
= 𝟏, 𝟎, 𝟎, 𝟎 , 𝟎, 𝟏, 𝟎, 𝟎 , 𝟎, 𝟎, 𝟏, 𝟎 , 𝟎, 𝟎, 𝟎, 𝟏
𝑻 𝒆 𝟏 = 𝟏, 𝟎, 𝟏, 𝟏
= 𝟏 𝟏, 𝟎, 𝟎, 𝟎 + 𝟎 𝟎, 𝟏, 𝟎, 𝟎 + 𝟏 𝟎, 𝟎, 𝟏, 𝟎 + 𝟏 𝟎, 𝟎, 𝟎, 𝟏

9. Cont…
𝑻 𝒆 𝟐 = 𝟏, 𝟏, 𝟎, 𝟎
= 𝟏 𝟏, 𝟎, 𝟎, 𝟎 + 𝟏 𝟎, 𝟏, 𝟎, 𝟎 + 𝟎 𝟎, 𝟎, 𝟏, 𝟎 + 𝟎 𝟎, 𝟎, 𝟎, 𝟏
𝑻 𝒆 𝟑 = 𝟎, 𝟏, −𝟏, 𝟎
= 𝟎 𝟏, 𝟎, 𝟎, 𝟎 + 𝟏 𝟎, 𝟏, 𝟎, 𝟎 − 𝟏 𝟎, 𝟎, 𝟏, 𝟎 + 𝟎 𝟎, 𝟎, 𝟎, 𝟏
The matrix A of T is A =
𝟏 𝟏 𝟎
𝟎 𝟏 𝟏
𝟏 𝟎 −𝟏
𝟏 𝟎 𝟎

10. Example:02
Find the matrix of each of the following linear
transformation with respect to the standard basis of the
given space.𝑻: 𝑹 𝟑 → 𝑹 𝟐 𝒅𝒆𝒇𝒊𝒏𝒆𝒅 𝒃𝒚 𝑻 𝒙 𝟏, 𝒙 𝟐, 𝒙 𝟑 =
𝟑𝒙 𝟏 − 𝟒𝒙 𝟐 + 𝟗𝒙 𝟑, 𝟓𝒙 𝟏 + 𝟑𝒙 𝟐 − 𝟐𝒙 𝟑
𝐬𝐭𝐚𝐧𝐝𝐚𝐫𝐝 𝐛𝐚𝐬𝐢𝐬 𝐨𝐟 𝑹 𝟑 = 𝟏, 𝟎, 𝟎 , 𝟎, 𝟏, 𝟎 , 𝟎, 𝟎, 𝟏
Standard basis of 𝑹 𝟐
= 𝟏, 𝟎 , 𝟎, 𝟏
𝐓 𝟏, 𝟎, 𝟎 = 𝟑, 𝟓 = 𝟑 𝟏, 𝟎 + 𝟓 𝟎, 𝟏
𝐓 𝟎, 𝟏, 𝟎 = −𝟒, 𝟑 = −𝟒 𝟏, 𝟎 + 𝟑 𝟎, 𝟏
𝐓 𝟎, 𝟎, 𝟏 = 𝟗, −𝟐 = 𝟗 𝟏, 𝟎 − 𝟐 𝟎, 𝟏)
𝐬𝐨 𝐭𝐡𝐞 𝐦𝐚𝐭𝐫𝐢𝐱 𝐨𝐟 𝐓 𝐢𝐬,
𝟑 −𝟒 𝟗
𝟓 𝟑 𝟐

11. example :03
Each of the following is the matrix of a linear
transformation𝐓: 𝐑 𝐧
→ 𝐑 𝐦
. Determine m, n and express T is
term of coordinates.
𝟑 𝟏 𝟎 𝟐 𝟎
𝟏 𝟎 𝟎 𝟏 𝟏
𝟎 −𝟏 𝟏 𝟏 𝟏
The order of matrix T is 3× 𝟓 .
n = 5, m = 3
𝑻: 𝑹 𝟓 → 𝑹 𝟑
𝐓
𝒙 𝟏
𝒙 𝟐
𝒙 𝟑
𝒙 𝟒
𝒙 𝟓
=
𝟑 𝟏 𝟎 𝟐 𝟎
𝟏 𝟎 𝟎 𝟏 𝟏
𝟎 −𝟏 𝟏 𝟏 𝟏
𝒙 𝟏
𝒙 𝟐
𝒙 𝟑
𝒙 𝟒
𝒙 𝟓

12. Cont…
=
𝟑𝒙 𝟏 + 𝒙 𝟐 + 𝟐𝒙 𝟒
𝒙 𝟏 + 𝒙 𝟒 + 𝒙 𝟓
𝒙 𝟏 − 𝒙 𝟐 + 𝒙 𝟑 + 𝒙 𝟒 + 𝒙 𝟓
𝐓 𝒙 𝟏, 𝒙 𝟐, 𝒙 𝟑, 𝒙 𝟒, 𝒙 𝟓
= 𝟑𝒙 𝟏 + 𝒙 𝟐 + 𝟐𝒙 𝟒, 𝒙 𝟏 + 𝒙 𝟒 + 𝒙 𝟓, 𝒙 𝟏 − 𝒙 𝟐 + 𝒙 𝟑
+ 𝒙 𝟒 + 𝒙 𝟓)