Merced College; Don Power

 

LINEAR ALGEBRA - CH 3-4, STUDY GUIDE

 

Chapter 3.  Given the points P(2,-1,3), Q(3,0,1), R(-1,0,3).

3-1.  Find the vectors v = PQ, w = PR, x=QR.

3-2.  Show that v + x = w.

3-3.  Vind a vector u = 3v + 2w.

3-4.  Find scalars c1 and c2 such that c1x + c2w = (1,5,8).

3-5.  Find |x| and show that is a unit vector.

3-6.  Find a vector of length 5 in the direction of v.

3-7.  Verify that |u+v| = |u| + |v|.

3-8.  Verify the Cauchy-Schwarz Inequality.

3-9.  Find the angle q (and cos q) at P in the triangle PQR.

3-10.  Prove that triangle OQR is a right triangle.

      a.  By finding the lengths of the sides and using Pythagoras.

      b.  By finding which two vectors (of OQ, OR, and QR) are orthogonal.

3-11.  Find projwv.  Does this vector project v onto w or does it project w onto v?

      Also find the length of projwv.

3-12.  Find the angle the vector v makes with the xz-plane.

3-13.  Find a vector orthogonal to both v and w.

3-14.  Use the fact that |u´v| = |u| |v| sin q to prove that |u´v| is the area of the parallelogram determined by u and v.

3-15.  Calculate i´j and i´k (for the standard unit vectors i, j, and k).

3-16.  Show that v, w, and x are coplanar using Thm 3.4.5.

3-17.  Find an equation of the plane containing P, Q, and R.

3-18.  Find parametric and/or vector equations of the line through P and R.

3-19.  Find the distance between (-2,2,7) and the plane in #17.

3-20.  Find the angle of intersection of the line r(t) = (5-2t, 3t,1+t) and the plane in #17.

Hints:  What is a vector parallel to the line? What is a normal vector to the plane (or any given plane)?

3-21.  Given a triangle determined by two vectors u and v, intersecting at an angle θ, relate the following vector expressions to parts of the triangle:  , , projuv, projvu, , , ,

 

Chapter 4

4-21.  Find the standard matrix of the linear transformation T given by w1 = 4x1 - 2x4, w2 = x1 -2x2 + x3  and    identify its domain and range.

4-22.  Calculate T(1,-2,3,1) for the transformation T in #21 by using

      a.  Direct substitution into the equations.

      b.  Multiplication by the matrix A of the linear transformation.

4-23.  Write the standard matrix A of a linear transformation T (easiest:  by considering the effect of the transformation on the standard unit vectors e1, e2, ...

      a.  Ex:  Write the standard matrix A for the orthogonal projection of vectors in R3 onto the yz-plane.

b.  Use multiplication by this matrix A to project (1,-2,3) onto the yz plane.

4-24.  Be ready with an example to show (a) geometrically and (b) by matrix multiplication, that the composition of linear transformations is not commutative.

4-25.  Multiply by the appropriate matrix to rotate the vector  counterclockwise through an angle of   p/3 (60 degrees).

4-26.    Given the standard matrix of T:Rn®Rn, determine whether

      a.  T is 1-1.

      b.  The range of T is all of Rn or only a subset of Rn.

4-27.  Given the standard matrix of a transformation T:Rn®Rn, find the image of any given standard basis vector (e.g. find T(e2)).

4-28.  Use Thm 4.3.2 to determine whether a given linear transformation is linear (See lesson 4.3, exercises 8-11).

4-29.  Find eigenvalues and eigenvectors for a geometric linear transformation, as in Example 8 of lesson 4.3 (see problems 18-19).  Check by calculating the eigenvalues and eigenvectors from the standard matrix of T.

 

Return to:  Merced College; Don Power               Updated 03/14/07 by Don Power