Sections 1.8/1.9: Linear Transformations

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Presentation transcript:

Sections 1.8/1.9: Linear Transformations

Recall that the difference between the matrix equation and the associated vector equation is just a matter of notation. However the matrix equation can arise is linear algebra (and applications) in a way that is not directly connected with linear combinations of vectors. This happens when we think of a matrix A as an object that acts on a vector by multiplication to produce a new vector

Example: A =

Recall that Ax is only defined if the number of columns of A equals the number of elements in the vector x.

A So multiplication by A transforms into .

In the previous example, solving the equation Ax = b can be thought of as finding all vectors x in R4 that are transformed into the vector b in R2 under the “action” of multiplication by A.

Transformation: Any function or mapping T Range Domain Codomain

Let A be an mxn matrix. Matrix Transformation: A Codomain Domain x b A

Example: The transformation T is defined by T(x)=Ax where For each of the following determine m and n.

Matrix Transformation: A x = b x A b Domain Codomain

Linear Transformation: Definition: A transformation T is linear if (i) T(u+v)=T(u)+T(v) for all u, v in the domain of T: (ii) T(cu)=cT(u) for all u and all scalars c. Theorem: If T is a linear transformation, then T(0)=0 and T(cu+dv)=cT(u)+dT(v) for all u, v and all scalars c, d.

Example. Suppose T is a linear transformation from R2 to R2 such that and . With no additional information, find a formula for the image of an arbitrary x in R2.

Theorem 10. Let be a linear transformation. Then there exists a unique matrix A such that for all x in Rn. In fact, A is the matrix whose jth column is the vector where is the jth column of the identity matrix in Rn. A is the standard matrix for the linear transformation T

Find the standard matrix of each of the following transformations. Reflection through the x-axis Reflection through the y-axis Reflection through the y=x Reflection through the y=-x Reflection through the origin

Find the standard matrix of each of the following transformations. Horizontal Contraction & Expansion Vertical Contraction & Expansion Projection onto the x-axis Projection onto the y-axis

Applets for transformations in R2 From Marc Renault’s collection… Transformation of Points http://webspace.ship.edu/msrenault/ggb/linear_transformations_points.html Visualizing Linear Transformations http://webspace.ship.edu/msrenault/ggb/visualizing_linear_transformations.html

Definition A mapping is said to be onto if each b in is the image of at least one x in . Definition A mapping is said to be one-to-one if each b in is the image of at most one x in . Theorem 11 Let be a linear transformation. Then, T is one-to-one iff has only the trivial solution. . Theorem 12 Let be a linear transformation with standard matrix A. 1. T is onto iff the columns of A span . 2. T is one-to-one iff the columns of A are linearly independent