Matrix of linear transformation

Matrix of linear transformation

• 2.
  FAZAIA COLLEGE OF EDUCATION FOR WOMEN PRESENTEDBY: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 operationof changing (as by rotation or mapping) one configuration or expression into another in accordance with a mathematical rule.
• 6.
  Linear transformation A lineartransformation 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 lineartransformation:  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 thematrix 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 matrixof 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 ofthe 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)