Section 1.9 The Matrix of a Linear Transformation Key Concepts Every linear transformation T :IRn IR m is actually a matrix transformation x Ax. • ! 7! The action of T on unit vectors e ,i=1...m, determines the structure of A. • i Important properties of T (one-to-one, onto) are intimately related to known properties of A. • 1 0 0 0 1 0 2 0 3 2 0 3 2 . 3 Example Given unit vectors e1 = , e2 = ,...,em = . , 6 . 7 6 . 7 6 7 6 . 7 6 . 7 6 0 7 6 7 6 7 6 7 6 0 7 6 0 7 6 1 7 6 7 6 7 6 7 any x IR m can be written as 4 5 4 5 4 5 2 x = x e + x e + + x e 1 1 2 2 ··· m m 10 0 x1 ···. 01 .. x2 = 2 3 2 3 . 6 . .. .. 0 7 . 6 7 6 7 6 0 017 6 xm 7 6 ··· 7 6 7 4 5 4 x 5 Im = |Imx. {z } | {z } Recall from section 1.8: if T :IRn IR m is a linear transformation, then ! T (cu + dv)=cT (u)+dT (v). General result: T (c v + c v + + c v )=c T (v )+c T (v )+ + c T (v ). 1 1 2 2 ··· p p 1 1 2 2 ··· p p Example Suppose T is a linear transformation from IR3 to IR4 with 2 5 1 3 0 −2 T (e )= ,T(e )= , and T (e )= . 1 2 −4 3 2 2 1 3 3 2 −0 3 6 5 7 6 1 7 6 7 7 6 7 6 − 7 6 7 4 5 4 5 4 5 x1 Compute T (x) for any x = x IR 3 and determine the matrix A that implements the transfor- 2 2 3 2 x3 mation T . 4 5 1 Question: What is the size of the matrix A that corresponds to the map T :IRp IR q? ! Theorem 10 Let T :IRn IR m be a linear transformation. Then there exists a unique m n matrix A such that ! ⇥ T (x)=Ax for all x in IRn. In fact, A is the m n matrix whose jth column is the vector T (e ), with e IR n: ⇥ j j 2 A =[T (e ) T (e ) T (e )] 1 2 ··· n The matrix A is called the standard matrix for the linear transformation T . Example Determine the standard matrices for the following linear transformations T :IR2 IR 2. ! Reflection across x1 axis Reflection across x2 axis Reflection across line x2 = x1 Reflection across line x = x 2 − 1 2 x1 2x2 Example Find the standard matrix for T :IR2 IR 3 if T : x 4−x . ! 7! 2 1 3 3x1 +2x2 4 5 Example Let T :IR2 IR 2 be the linear transformation that rotates each point in IR2 about the origin through and angle! ⇡/4 radians (counterclockwise). Determine the standard matrix for T . Question: Determine the standard matrix for the linear transformation T :IR2 IR 2 that rotates each point in IR2 counterclockwise around the origin through an angle of φ radians.! 3 Definitions: A mapping T :IRn IR m is onto IR m if each b in IRm is the image of at least one x in IRn. • ! Equivalent ways to state this definition are: Note: T is not onto IRm when there is some b in IRm for which the equation T (x)=b has no solution, or equivalently, whenever the is some b in IRm for which the system [A b] is inconsis- tent. | A mapping T :IRn IR m is one to one if each b in IRm is the image of at most one x in IRn. • ! Example Let T be the transformation whose standard matrix is 13 41 A = 02−30 . 2 00− 033 4 5 Does T map IR4 onto IR3? Is T a one-to-one mapping? Explain your reasoning. 4 Theorem 11 Let T :IRn IR m be a linear transformation. Then T is one-to-one if and only if the equation T (x)=0 has only! the trivial solution. Proof: Theorem 12 Let T :IRn IR m be a linear transformation and let A be the standard matrix for T . Then: ! a. T maps IRn onto IRm if and only if the columns of A span IRm. b. T is one-to-one if and only if the columns of A are linearly independent. Proof: 5 Summary of section 1.9 Each T :IRn IR m corresponds to standard matrix A of size m n. • ! ⇥ The standard matrix is determined by the action of T on the standard unit vectors e . • i Properties of T (onto, one-to-one) are deeply related to questions of existence/uniqueness of • solutions to Ax = b. 6.