Sol:Elementary Linear Algebra: Section 6.2, pp.379-380 Notes:Elementary Linear Algebra: Section 6.2, p.379Р”В®x 7: (Finding a basis for the range of a linear transformation)Find a basis for the range of T.Elementary Linear Algebra: Section 6.2, p.379

$onto vs one to one matrix$

Thm 6.4: (The range of T is a subspace of W)Pf:Elementary Linear Algebra: Section 6.2, p.379 Sol:Elementary Linear Algebra: Section 6.2, p.378Р”В¬orollary to Thm 6.3:Range of a linear transformation T:Elementary Linear Algebra: Section 6.2, p.378

Elementary Linear Algebra: Section 6.2, p.377Р”В®x 6: (Finding a basis for the kernel)Find a basis for ker(T) as a subspace of R5.Elementary Linear Algebra: Section 6.2, p.377 Thm 6.3: (The kernel is a subspace of V)The kernel of a linear transformation is a subspace of the domain V.Pf:Note:The kernel of T is sometimes called the nullspace of T. Keywords in Section 6.1:function: domain: codomain: image of v under T: Tvrange of T: Tpreimage of w: wlinear transformation: linear operator: zero transformation: identity transformation:Р”В¦.2 The Kernel and Range of a Linear TransformationKernel of a linear transformation T:Let be a linear transformationThen the set of all vectors v in V that satisfy is called the kernel of T and is denoted by ker(T).Elementary Linear Algebra: Section 6.2, p.375Р”В®x 2: (The kernel of the zero and identity transformations)(a) T(v)=0 (the zero transformation )(b) T(v)=v (the identity transformation )Elementary Linear Algebra: Section 6.2, p.375Р”В®x 5: (Finding the kernel of a linear transformation)Sol:Elementary Linear Algebra: Section 6.2, p.376Р”В®lementary Linear Algebra: Section 6.2, p.377 Elementary Linear Algebra: Section 6.1, p.368Р”В®x 8: (A projection in R3)is called a projection in R3.The linear transformation is given byElementary Linear Algebra: Section 6.1, p.369Р”В®x 9: (A linear transformation from Mmn into Mn m )Show that T is a linear transformation.Sol:Therefore, T is a linear transformation from Mmn into Mn m.Elementary Linear Algebra: Section 6.1, p.369 Rthe length of T(v) +the angle from the positive x-axis counterclockwise to the vector T(v)Thus, T(v) is the vector that results from rotating the vector v counterclockwise through the angle. Sol:(polar coordinates) r the length of vthe angle from the positive x-axis counterclockwise to the vector v Elementary Linear Algebra: Section 6.1, p.368 given by the matrixhas the property that it rotates every vector in R2 counterclockwise about the origin through the angle. The function T defined byis a linear transformation from Rn into Rm.Note:Elementary Linear Algebra: Section 6.1, p.367Р”В®x 7: (Rotation in the plane)Show that the L.T. Thm 6.2: (The linear transformation given by a matrix)Let A be an mn matrix. Zero transformation:Identity transformation:Thm 6.1: (Properties of linear transformations)Elementary Linear Algebra: Section 6.1, p.365Р”В®x 4: (Linear transformations and bases)Let be a linear transformation such that Sol:(T is a L.T.)Find T(2, 3, -2).Elementary Linear Algebra: Section 6.1, p.365Р”В®x 5: (A linear transformation defined by a matrix)The function is defined asSol:(vector addition)(scalar multiplication)Elementary Linear Algebra: Section 6.1, p.366 (2) is not a linear transformation from a vector space R into R because it preserves neither vector addition nor scalar multiplication.Elementary Linear Algebra: Section 6.1, p.364 Notes: Two uses of the term linear.(1) is called a linear function because its graph is a line. Therefore, T is a linear transformation.Elementary Linear Algebra: Section 6.1, p.363Р”В®x 3: (Functions that are not linear transformations)Elementary Linear Algebra: Section 6.1, p.363 Notes:(1) A linear transformation is said to be operation preserving.(2) A linear transformation from a vector space into itself is called a linear operator.Elementary Linear Algebra: Section 6.1, p.363Р”В®x 2: (Verifying a linear transformation T from R2 into R2)Pf:Elementary Linear Algebra: Section 6.1, p.363 Linear Transformation (L.T.):Elementary Linear Algebra: Section 6.1, p.362 (b) Find the preimage of w=(-1,11)Sol:Thus is the preimage of w=(-1, 11).Elementary Linear Algebra: Section 6.1, p.362 the range of T: The set of all images of vectors in V.the preimage of w: The set of all v in V such that T(v)=w.Elementary Linear Algebra: Section 6.1, p.361Р”В®x 1: (A function from R2 into R2 )(a) Find the image of v=(-1,2). Image of v under T:If v is in V and w is in W such thatThen w is called the image of v under T. (6 Edition)Р”В¦.1 Introduction to Linear TransformationsFunction T that maps a vector space V into a vector space W:V: the domain of TW: the codomain of TElementary Linear Algebra: Section 6.1, pp.361-362 Chapter 6Linear Transformations6.1 Introduction to Linear Transformations6.2 The Kernel and Range of a Linear Transformation6.3 Matrices for Linear Transformations6.4 Transition Matrices and SimilarityElementary Linear AlgebraR.

Onto vs one to one matrix