Linear Algebra (wi1403lr) Lecture no.4
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1 Linear Algebra (wi1403lr) Lecture no.4 EWI / DIAM / Numerical Analysis group Matthias Möller 29/04/2014 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
2 Review of lecture no Solution Sets of Linear Systems homogeneous linear systems always have trivial solution have nontrivial solutions if and only if there are free variables inhomogeneous linear systems solution=particular solution+homogeneous solution parametric form of solutions: (translated) lines, planes, etc. geometric interpretation of solution sets M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
3 Review of lecture no.3, cont d 1.7 Linear Independence definition relation between linear independence of matrix columns and existence of nontrivial solutions to the matrix equation criteria for linear dependence of indexed set {v 1,..., v p } v j can be written as linear combination of v 1,..., v j 1 set contains more vectors than there are entries in each vector set contains the zero vector M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
4 Learning objectives of lecture no.4 You will learn to describe manipulations of vectors by transformations to derive and analyse linear matrix transformations to identify different types of mappings/transformations M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
5 Describe the shade mathematically! M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
6 Describe the shade mathematically! aircraft shade M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
7 Formal description... The aircraft is a three-dimensional object that should be mapped to the two-dimensional ground M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
8 Formal description... The aircraft is a three-dimensional object (domain of T ) that should be mapped (i.e. transformed by mapping T ) to the two-dimensional ground (codomain of T ). T x R 3 Domain R 2 Codomain Range T (x) M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
9 ...in words Transformation A transformation T from R n to R m is a rule that assigns to each vector x R n a vector T (x) R m. We use the notation T : R n R m The set R n is called the domain of T, and R m is called the codomain of T. For a particular x R n, the vector T (x) R m is called the image of x under the action of the transformation T. The set of all images T (x) is called the range of T. M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
10 Matrix transformation Let transformation T : R n R m be defined as [ ] T (x) := x, x = Give values for domain and codomain, aka n and m. 1 2 Compute the image b = T (x) of the vector x = 2. 3 x 2 x 3 3 Find vector x whose image is b = [ ] 0 4 Is vector x = in the range of T? 0 x 1 x 2 x 3 [ ] 2. Is it unique? 2 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
11 Matrix transformation Let transformation T : R n R m be defined as [ ] T (x) := x, x = Give values for domain and codomain, aka n and m. 1 2 Compute the image b = T (x) of the vector x = 2. 3 x 2 x 3 3 Find vector x whose image is b = [ ] 0 4 Is vector x = in the range of T? 0 x 1 x 2 x 3 [ ] 2. Is it unique? 2 See EXAMPLE 1 for another worked out example. M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
12 Matrix transformation Take home lesson 1 Finding the image b (in the codomain) of a vector x (in the domain) amounts to multiplying it with the matrix A x b = Ax 2 Finding all vectors x (in the domain) that are transformed into the image vector b (in the codomain) under the multiplication by matrix A amounts to solving the equation Ax = b. M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
13 Matrix transformation Quiz: Let T (x) := Ax with n m transformation matrix A. Then the domain of T is the codomain of T is the range of T is M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
14 Matrix transformation Quiz: Let T (x) := Ax with n m transformation matrix A. Then the domain of T is R n since A has n columns the codomain of T is the range of T is M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
15 Matrix transformation Quiz: Let T (x) := Ax with n m transformation matrix A. Then the domain of T is R n since A has n columns the codomain of T is R m since A has m rows the range of T is M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
16 Matrix transformation Quiz: Let T (x) := Ax with n m transformation matrix A. Then the domain of T is R n since A has n columns the codomain of T is R m since A has m rows the range of T is the set of all linear combinations of the columns of A M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
17 Special matrix transformation The identity transformation x x maps each vector x in R 3 to x in R 3. What is the image of x = x 2? x 3 x 1 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
18 Special matrix transformation The scaled identity transformation x k x scales each vector x in R 3 to kx in R 3. What is the image of x = x 2 with k = 2? x 3 x 1 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
19 Special matrix transformation The projection transformation x x projects each vector x in R 3 onto the x 1 x 3 -plane. What is the image of x = x 2? x 3 x 1 What is the transformation matrix for our aircraft example? M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
20 Special matrix transformation The horizontal contraction/expansion transformation [ ] [ ] k 0 kx1 x x = 0 1 contracts (0 < k < 1) or expands (k > 1) each vector x = horizontally. x 2 [ x1 x 2 ] What is the range of the square [0, 0] [1, 1] under T? M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
21 Special matrix transformation The horizontal contraction/expansion transformation [ ] [ ] k 0 kx1 x x = 0 1 contracts (0 < k < 1) or expands (k > 1) each vector x = horizontally. x 2 [ x1 x 2 ] What is the range of the square [0, 0] [1, 1] under T? A similar transformation exists in vertical direction. M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
22 Special matrix transformation The horizontal shear transformation [ ] [ ] 1 k x1 + kx x x = deforms each vector x = [ x1 x 2 x 2 ], dragging its x 1 -component to the right (k > 0) and to the left (k < 0), respectively. What is the range of the square [0, 0] [1, 1] under T? M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
23 Special matrix transformation The horizontal shear transformation [ ] [ ] 1 k x1 + kx x x = deforms each vector x = [ x1 x 2 x 2 ], dragging its x 1 -component to the right (k > 0) and to the left (k < 0), respectively. What is the range of the square [0, 0] [1, 1] under T? A similar transformation exists in vertical direction. M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
24 Special matrix transformation The reflection transformation [ ] 0 1 x x = 1 0 reflects each vector x = [ x1 x 2 [ x2 x 1 ] ] along the line x 2 = x 1 What is the range of the square [0, 0] [1, 1] under T? M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
25 Special matrix transformation The rotation transformation [ ] cos φ sin φ x x sin φ cos φ rotates each vector x = [ x1 x 2 ] about the origin through angle φ with counterclockwise orientation for positive angle φ. What is the range of the square [0, 0] [1, 1] under T for φ = π/2 and φ = π? M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
26 Special matrix transformation The rotation transformation [ ] cos φ sin φ x x sin φ cos φ rotates each vector x = [ x1 x 2 ] about the origin through angle φ with counterclockwise orientation for positive angle φ. What is the range of the square [0, 0] [1, 1] under T for φ = π/2 and φ = π? More examples on pages M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
27 Linear transformation It follows from the properties of the matrix-vector product that A(u + v) = Au + Av A(cu) = cau This leads to the following for all u, v in R n for all u in R n and scalars c R Definition The arbitrary transformation T is linear if T (u + v) = T u + T v T (cu) = ct u for all u, v in the domain of T for all u in the domain of T and scalars c R Every matrix transformation is linear. Why? M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
28 Properties of linear transformation Quiz: True or false (then give a counterexample) Let T be a linear transformation. Then T (0) = 0 True False T (au + bv) = at (u) + bt (v) True False for all vectors u, v in the domain of T and all scalars a, b the image {T (v 1 ), T (v 2 ), T (v 3 )} of a linearly dependent set {v 1, v 2, v 3 } is linearly dependent, too True False M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
29 Properties of linear transformation Quiz: True or false (then give a counterexample) Let T be a linear transformation. Then T (0) = 0 True False T (au + bv) = at (u) + bt (v) True False for all vectors u, v in the domain of T and all scalars a, b the image {T (v 1 ), T (v 2 ), T (v 3 )} of a linearly dependent set {v 1, v 2, v 3 } is linearly dependent, too True False M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
30 Properties of linear transformation Quiz: True or false (then give a counterexample) Let T be a linear transformation. Then T (0) = 0 True False T (au + bv) = at (u) + bt (v) True False for all vectors u, v in the domain of T and all scalars a, b the image {T (v 1 ), T (v 2 ), T (v 3 )} of a linearly dependent set {v 1, v 2, v 3 } is linearly dependent, too True False M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
31 Properties of linear transformation Quiz: True or false (then give a counterexample) Let T be a linear transformation. Then T (0) = 0 True False T (au + bv) = at (u) + bt (v) True False for all vectors u, v in the domain of T and all scalars a, b the image {T (v 1 ), T (v 2 ), T (v 3 )} of a linearly dependent set {v 1, v 2, v 3 } is linearly dependent, too True False M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
32 Linear = matrix transformation? Every matrix transformation is linear due to the properties of matrix-vector multiplication. There are transformations which are not linear, e.g. ([ ]) [ ] x1 x1 2 x T := 2 x 2 x 1 4x 2 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
33 Linear = matrix transformation? Every matrix transformation is linear due to the properties of matrix-vector multiplication. There are transformations which are not linear, e.g. ([ ]) [ ] x1 x1 2 x T := 2 x 2 x 1 4x 2 T ([ ]) 1 + T 1 ([ ]) 1 = 1 [ ] [ ] 3 T 3 ([ ]) 0 = 0 Is there a matrix representation of this transformation? [ ] 0 0 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
34 Linear = matrix transformation? Every matrix transformation is linear due to the properties of matrix-vector multiplication. There are transformations which are not linear, e.g. ([ ]) [ ] x1 x1 2 x T := 2 x 2 x 1 4x 2 T ([ ]) 1 + T 1 ([ ]) 1 = 1 [ ] [ ] 3 T 3 ([ ]) 0 = 0 Is there a matrix representation of this transformation? No! [ ] 0 0 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
35 Linear = matrix transformation! Let T : R n R m be a linear transformation. Then there exists a unique matrix A such that T (x) = Ax for all x in R n A is the m n matrix whose jth column is the vector T (e j ), where e j is the jth column of the identity matrix in R n. The matrix A = [T (e 1 ),..., T (e n )] is called the standard matrix for the linear transformation. M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
36 Finding the standard matrix Find the standard matrix for the mapping T : R 4 R 4 with x 1 + 2x 2 T (x 1, x 2, x 3, x 4 ) = 0 2x 2 + x 4 x 2 x 4 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
37 Finding the standard matrix Find the standard matrix for the mapping T : R 4 R 4 with x 1 + 2x x 1 T (x 1, x 2, x 3, x 4 ) = 0 2x 2 + x 4 = x x 3 x 2 x x 4 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
38 Finding the standard matrix Find the standard matrix for the mapping T : R 2 R 2 that 1 performs a horizontal shear that maps e 1 into e 2 + 2e 1 (leaving e 1 unchanged) 2 and then reflects points through the line x 2 = x 1 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
39 Finding the standard matrix Find the standard matrix for the mapping T : R 2 R 2 that 1 performs a horizontal shear that maps e 1 into e 2 + 2e 1 (leaving e 1 unchanged) [ ] [ ] [ ] [ ] 1 k x1 1 k 0 T 1 (x) := = x 2 [ ] k 1! = [ ] 2 k = and then reflects points through the line x 2 = x 1 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
40 Finding the standard matrix Find the standard matrix for the mapping T : R 2 R 2 that 1 performs a horizontal shear that maps e 1 into e 2 + 2e 1 (leaving e 1 unchanged) [ ] [ ] [ ] [ ] 1 k x1 1 k 0 T 1 (x) := = x 2 [ ] k 1! = [ ] 2 k = and then reflects points through the line x 2 = x 1 [ ] [ ] 0 1 x1 T 2 (x) := cf. page x 2 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
41 Finding the standard matrix Find the standard matrix for the mapping T : R 2 R 2 that 1 performs a horizontal shear that maps e 1 into e 2 + 2e 1 (leaving e 1 unchanged) [ ] [ ] [ ] [ ] 1 k x1 1 k 0 T 1 (x) := = x 2 [ ] k 1! = [ ] 2 k = and then reflects points through the line x 2 = x 1 [ ] [ ] 0 1 x1 T 2 (x) := cf. page Solution: Concatenate both transformations by multiplication [ ] ([ ] [ ]) [ ] [ ] x1 0 1 x1 T (x) = T 2 (T 1 (x)) := = x 2 x 2 x 2 M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
42 Types of mappings A mapping T : R n R m is said to be onto R m if each b in R m is the image of at least one x in R n. mapping T reaches all locations in R m A mapping T : R n R m is said to be one-to-one if each b in R m is the image of at most one x in R n. no two vectors in R n have the same image M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
43 Types of mappings Quiz: True or false (then give a counterexample) The linear mapping x 1 + 2x x 1 T (x 1, x 2, x 3, x 4 ) = 0 2x 2 + x 4 = x x 3 x 2 x x 4 maps R 4 onto R 4 True False is a one-to-one mapping True False M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
44 Types of mappings Quiz: True or false (then give a counterexample) The linear mapping x 1 + 2x x 1 T (x 1, x 2, x 3, x 4 ) = 0 2x 2 + x 4 = x x 3 x 2 x x 4 maps R 4 onto R 4 True False 0 There is no vector in R 4 that is mapped to is a one-to-one mapping True False M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
45 Types of mappings Quiz: True or false (then give a counterexample) The linear mapping x 1 + 2x x 1 T (x 1, x 2, x 3, x 4 ) = 0 2x 2 + x 4 = x x 3 x 2 x x 4 maps R 4 onto R 4 True False 0 There is no vector in R 4 that is mapped to is a one-to-one mapping True False M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
46 Types of mappings Let T : R n R 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. Let T : R n R m be a linear transformation and let A be the standard matrix for T. Then 1 T maps R n onto R m if and only if the columns of A span R m ; 2 T is one-to-one if and only if the columns of A are linearly independent. M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
47 Types of mappings Quiz: True or false (then give a counterexample) The linear mapping x 1 + 2x x 1 T (x 1, x 2, x 3, x 4 ) = 0 2x 2 + x 4 = x x 3 x 2 x x 4 maps R 4 onto R 4 True False is a one-to-one mapping True False M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
48 Types of mappings Quiz: True or false (then give a counterexample) The linear mapping x 1 + 2x x 1 T (x 1, x 2, x 3, x 4 ) = 0 2x 2 + x 4 = x x 3 x 2 x x 4 maps R 4 onto R True False The columns span a b c 0 1 with scalars a, b, c is not equal to R is a one-to-one mapping True False M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
49 Types of mappings Quiz: True or false (then give a counterexample) The linear mapping x 1 + 2x x 1 T (x 1, x 2, x 3, x 4 ) = 0 2x 2 + x 4 = x x 3 x 2 x x 4 maps R 4 onto R True False The columns span a b c 0 1 with scalars a, b, c is not equal to R is a one-to-one mapping True False 3 columns with 2 entries each cannot be linearly independent M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
50 Types of mappings Quiz: True or false (then give a counterexample) The linear mapping [ ] x1 5x T (x 1, x 2, x 3 ) = 2 + 4x 3 = x 2 6x 3 [ ] 1 x x 2 x 3 maps R 3 onto R 2 True False is a one-to-one mapping True False M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
51 Types of mappings Quiz: True or false (then give a counterexample) The linear mapping [ ] x1 5x T (x 1, x 2, x 3 ) = 2 + 4x 3 = x 2 6x 3 [ ] 1 x x 2 x 3 maps R 3 onto R 2 True False [ ] [ ] [ ] The columns span a + b + c with scalars a, b, c which is equal to R 2 is a one-to-one mapping True False M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
52 Types of mappings Quiz: True or false (then give a counterexample) The linear mapping [ ] x1 5x T (x 1, x 2, x 3 ) = 2 + 4x 3 = x 2 6x 3 [ ] 1 x x 2 x 3 maps R 3 onto R 2 True False [ ] [ ] [ ] The columns span a + b + c with scalars a, b, c which is equal to R 2 is a one-to-one mapping True False Three columns with two entries each cannot be linearly independent M. Möller (EWI/NA group) LA (wi1403lr) 29/04/ / 28
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