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Understand Linear Transformations in Algebra

Learn the concept of linear transformations, matrix equations, and applications in linear algebra. Explore matrix transformations, standard matrices, and theorems related to linear maps. Use interactive applets for visualizing transformations in R2.

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Understand Linear Transformations in Algebra

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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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