Linear Independence, Linear Transformation Math 112, Week 3

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Linear independence, Linear transformation Math 112, Week 3

Goals:

• Solutions of linear system in parametric vector form.

• Linear independence and linear dependence.

• Introduction to Linear transformation.

Suggested Textbook Readings: Sections §1.5, §1.7, §1.8, §1.9

Steps of writing the solution set (of a consistent system) in parametric vector form:

1. Row reduce the augmented matrix to RREF.

2. Express each basic variable in terms of any free variables appearing in an equation.

3. Write a typical solution ~x as a vector whose entries depend on the free variables, if any.

4. Decompose ~x into a linear combination of vectors (with numeric entries) using the free variables as parameters.

Facts of linearly dependent set of vectors:

1. A set S = {~v1, ··· ,~vk} of two or more vectors is linearly dependent if and only if at least one of the vectors in S is a linear combination of the others.

2. If a set contains more vectors than there are entries in each vector, then the set n is linearly dependent. That is, any set {~v1, ··· ,~vk} in R is linearly dependent if k > n.

n 3. If a set S = {~v1, ··· ,~vk} in R contains the zero vector, then the set is linearly dependent. Week 3: Linear independence, Linear transformation 2

Example 1: Solve the system and write the solutions in vector form.

 3x + 5x − 4x = 0  1 2 3 −3x1 − 2x2 + 4x3 = 0   6x1 + x2 − 8x3 = 0

Example 2: Solve the system and write the solutions in vector form.

2x1 − 3x2 − 4x3 = 0

MATH 112 Winter 2016 Week 3: Linear independence, Linear transformation 3

Example 3: Solve the system and write the solutions in vector form.

 3x + 5x − 4x = 3  1 2 3 −3x1 − 2x2 + 4x3 = 0   6x1 + x2 − 8x3 = −3

Example 4: Solve the system and write the solutions in vector form.

2x1 − 3x2 − 4x3 = 1

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Example 5: Determine if the following sets of vectors are linearly indepen- dent.         5 2 1 −1 1. {  ,   ,   ,  } 1 8 3 4

    4 6         2. {−2 , −3}     6 9

      3 0 −6             3. { 5  , 0 ,  5 }       −1 0 4

    −8 2         4. { 12  , −3}     −4 −1

      1 2 3             5. {2 , 3 , 5}       3 4 7

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Matrix transformation and Linear Transformation

Definition: Let A be an m × n matrix. The transformation T : Rn → Rm given by T (~x) = A~x is called a matrix transformation.   1 1 For example, T (~x) = A~x, where A =  , is a matrix transformation from R2 to 1 −1   1 2 −1 R2, and the matrix B =   defines a transformation T : R3 → R2, T (~x) = B~x 1 0 3 for ~x ∈ R3. Definition: A transformation for which

T (~u + ~v) = T (~u) + T (~v), and T (k~u) = kT (~u) where k is a scalar, is called a linear transformation.     x1 2x1 For example, T : R2 → R2, with T ( ) =   is a linear transformation, but x2 x2     2 x1 x1 + x2 T ( ) =   is not a linear transformation. x2 x2

Theorem:Every matrix transformation is a linear transformation.

Theorem:Every linear transformation T : Rn → Rm has a unique matrix A so that T (~x) = A~x. This matrix is called the standard matrix of T .

Standard matrix of a linear transformation: The standard matrix of a linear n m transformation T : R → R can be found by computing T (~ej), j = 1, 2, ··· , n, where n ~ej denotes the vector in R whose j-th entry is 1 and the rest entries are 0. The standard matrix of T is given by: h i T (~e1) T (~e2) ··· T (~en)

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  1 −3     Example 6: Let A =  3 5 , and deﬁne a transformation   −1 7

T : R2 → R3 by T (~x) = A~x.   2 1. Let ~u =  , ﬁnd T (~u), the image of ~u under T . −1

  3   ~   2 ~ 2. Let b =  2 , ﬁnd an ~x in R whose image under T is b.   −5

3. Is there more than one ~x whose image under T is ~b?

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  1 −3     A =  3 5 , T (~x) = A~x   −1 7   3     4. Let ~c = 2, determine if ~c is in the range of T .   5

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Example 7: (Rotation Transformation) Let T : R2 → R2 be the transformation that rotates each point in R2 about the origin through an angle θ, with counterclockwise rotation for a positive angle. This is a linear transformation. Find its standard matrix.

MATH 112 Winter 2016