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Lecture 15

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MATH 136
Robert Sproule

Friday, February 7 − Lecture 15 : Linear mappings (transformations) II Concepts: 1. matrix induced by a linear mapping. 2. Examples of linear mappings. n m 15.1 Theorem − Let T : ℝ → ℝ be a linear mapping. Then there exists a unique m × n matrix A such that Ax = T(x) for all x ∈ ℝ . n Remark − We see that x and Ax are viewed as column matrices and thatx and T(x) are n m viewed simultaneously as vectors in ℝ and ℝ respectively. - It will often be useful to equate a column matrix to a vector when the corresponding entries are the same. - By the equation Ax = T(x) we will mean that "the entries of the column matrix Ax are identical to the respective entries of the vector T(x)". n m Proof of the theorem: Given : T : ℝ → ℝ be a linear mapping. We claim: There exists at least one matrix A m × n such that Ax = T(x). Proof of claim : Let w = T(e) for i = 1 to n. (The vector e is the vector whose i i th i n components are zeros, except for the i component, which is a 1.) Note that w ∈ ℝ for i i = 1 to n. We define the matrix,A m × n= [w w1… w2] n m × n, as being the m × n matrix whose columns are the w's.iNote that Ae = w 1 Ae =1w , …2Ae = 2 = T(e) ior eaih i = 1ito n. n - If x ∈ ℝ , then x = ∑ i = 1 to ni ir some scalars α sinci {e} formi a basis of ℝ n. Hence as claimed. We claim: Such a matrix A associated to the linear mapping T in this way is unique. Proof of claim : Suppose B is another matrix such that Bx = T(x) for all x ∈ ℝ . n Then, if we let C = A − B, Cx = (A − B)x = Ax − Bx = T(x) − T(x) = 0 n Tthn C is a matrix such that Cx = 0 for all x in ℝ . Thus Ce = 0 foriall e, But Ce i i i column of C. Thus C = 0-matrix. It follows that A = B. So the matrix A is unique. 15.1.1 Remarks. n m 1. The unique matrix A associated to the linear transformation T : ℝ → ℝ such that Ax = T(x) will be referred to as the “matrix A m×n induced by T ” or the “standard matrix of T ”. n m 2. Given, a linear matrix T : ℝ → ℝ we can associate a unique m× n matrix A whose columns are T(e ), T(1 ), …,2T(e ). We hnve shown that T(x) = Ax. 15.2 Simple examples of linear mappings. 2 2 1) Reflection into the x-axis. Consider the matrix A:ℝ → ℝ : Viewed as a linear transformation it will map vectors in ℝ 2to vectors in ℝ .2 - We see that - - We see that this particular matrix "reflects" vectors in ℝ about the x-axis. - This i
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