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Sections 1.8/1.9: Linear Transformations

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 .

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Sections 1.8/1.9: Linear Transformations

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  1. Sections 1.8/1.9: Linear Transformations

  2. 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

  3. Example: A =

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

  5. A So multiplication by A transforms into .

  6. In the previous example, solving the equation Ax = b can be thought of as finding all vectors x in R4that are transformed into the vector b in R2 under the “action” of multiplication by A.

  7. Transformation: Any function or mapping T Range Domain Codomain

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

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

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

  11. 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.

  12. 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.

  13. Theorem 10. Let be a linear transformation. Then there exists a unique matrix A such that for all xin 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

  14. 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

  15. 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

  16. 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

  17. 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

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