Category Archives: linear algebra

Exercise 2.6.8. If is the space of cubic polynomials in , what matrix would represent ? What are the nullspace and column space of this matrix? What polynomials would they represent? Answer: consists of all polynomials of the form . … Continue reading →

Exercise 2.6.7. Describe the matrices representing the following transformations: i) projecting all vectors onto the – plane ii) reflecting all vectors through the – plane iii) rotating all vectors in the – plane by 90 degrees, leaving the axis unchanged … Continue reading →

Exercise 2.6.6. Suppose we have a transformation matrix This is a shearing transformation: it leaves unchanged any points for which and thus leaves unchanged the entire -axis. Describe how this transformation affects the -axis, including the points , , and … Continue reading →

Exercise 2.6.5. Suppose we have two points and and a third point halfway between the first two points. Show that for any linear transformation represented by a matrix the (transformed) point is halfway between and . Answer: If is halfway … Continue reading →

Exercise 2.6.4. Consider the matrix This matrix will “stretch” vectors along the x axis, transforming the vector into the vector . Consider also the circle formed by all points for which . What shape is the curve created by transforming … Continue reading →

Exercise 2.6.3. Suppose we form the product of  2 by 2 matrices representing 5 reflections and 8 rotations. Does that product matrix represent a reflection or a rotation? Answer: I’ll show a long way to the answer and then a … Continue reading →

This post summarizes the results of previous posts exploring the effects of the following sequences of linear transformations in the x-y plane: a rotation followed by a rotation a reflection followed by a reflection a rotation followed by a reflection … Continue reading →

In preparation for answering exercise 2.6.3 in Gilbert Strang’s Linear Algebra and Its Applications, Third Edition, I wanted to derive in detail the effect of a rotation followed by a rotation, a reflection followed by a reflection, a rotation followed … Continue reading →

In preparation for answering exercise 2.6.3 in Gilbert Strang’s Linear Algebra and Its Applications, Third Edition, I wanted to derive in detail the effect of a rotation followed by a rotation, a reflection followed by a reflection, a reflection followed … Continue reading →

In preparation for answering exercise 2.6.3 in Gilbert Strang’s Linear Algebra and Its Applications, Third Edition, I wanted to derive in detail the effect of a reflection followed by a reflection, a reflection followed by a rotation, and a rotation … Continue reading →