Marywood University Department of Mathematics Course Information for Math 322 Linear Algebra by Dr. J.

Carl Friedrich Gauss (1777-1855)

An introduction to Linear Algebra. This course is required for both mathematics and math/secondary education majors. Topics include linear systems, matrices, determinants, vector spaces, dimension, linear transformations, eigenvectors and eigenvalues, norms, and orthogonality. The textbook is Linear Algebra with Maple by Fred Szabo and published by Harcourt and Academic Press. The course grade is determined by three tests, six or seven Maple Labs, and a final examination. Useful websites are listed at the end of this page. Prerequisite: Math 202 (Calculus II).

Sample Problems for Test #1

• True or false
  • The diagonal elements of a symmetric matrix must all equal 0.
  • For a square matrix A, if Ax = 0 has a nontrivial solution, then A is not invertible.
  • If m > n for an m x n homogeneous system of linear equations, then there must be an infinite number of solutions.
  • For any size matrix A, it is true that A^TA is a square matrix.
  • If A and B are square matrices, then det(A + B) = det A + det B.
  • The linear transformation T: Rⁿ --> Rⁿ defined by T(x) = Ax for an n x n matrix A is one-to-one if and only if det A ‚ 0.
  • Row equivalent matrices always have the same determinant.
  • For any square matrices A, B, of the same size, (A + B)² = A² + 2AB + B² .
• Given the three matrices A, B, C, perform the following operations. (These matrices are not listed here due to the complexity of forming them with HTML.) The computation may be undefined. If that is the case, then simply state that it is undefined

  (i) 3A + 2B (ii) 3A + 2C (iii) AB (iv) CA (v) det A

• Determine whether the matrix A is invertible and, if it is, find A^-1. Use the row reduction method and show all your intermediate matrices. (Please use fractions and not decimals.)
• Use the Gaussian elimination method on augmented matrices to solve the following systems.

  (i) 4x + 8y - 12z = 44

  3x + 6y - 8z = 32

  -2x - y = -7

  (ii) 3x - 5y + z = 0

  - x - y + z = - 4

  2x - 4y + z = -1

• If A is any square matrix, show that the matrix B = A + A^T is symmetric.
• If A is an orthogonal matrix, then prove that det A = ± 1.
• Prove: Any rotation matrix is an orthogonal matrix.
• Prove: If P and Q are n x n orthogonal matrices, then PQ is also orthogonal.
• Use Cramer's Rule to solve for x and y in the following system.

  -2x + 3y - z = 1

  x + 2y - z = 4

  -2x - y + z = -3

Return to top

Sample Problems for Test #2

• 1. True/False.
  • If v is an element of span{v₁, v₂, . . . , v_k} , then {v₁, v₂, . . . , v_k, v} must be linearly dependent.
  • A finite dimensional vector space cannot have an infinite dimensional subspace.
  • If A is an n x n matrix that is row-reducible to the n x n identity matrix I, then nullity(A) = 1.
  • If dim V = n, then any set with more than n vectors must be linearly dependent.
  • If dim V = n, then any set with less than n vectors must be linearly independent.
  • The hyperplane H = { (x, y, z, w) | ax + by + cz + dw = 0 } is a subspace of R⁴ such that dim H = 4.
  • rank(A) = rank(A^T) for any m x n matrix A.
  • For any matrix A, dim(Col A) = n - dim(Row A).
  • If A is an m x n matrix and nullity(A) > 0, then rank(A) < n.
  • A set of n vectors in R^m is always linearly dependent if n > m.
• Which of the following subsets H of the given vector space V is a subspace of V? If it is not a subspace, explain why not. If it is a subspace, prove it.

  (i) V = R² ; H = { (x, y) | x² + y² = 1} (ii) V = C[0, 1] ; H = { f is an element of C[0, 1] with integral 0 over [0,1]}

• Let { v₁, v₂, v₃ } be a linearly independent set. If u₁ = v₁, u₂ = v₁ + v₂ , and u₃ = v₁ + v₂ + v₃, show that the set { u₁, u₂, u₃ } is also linearly independent.
• Given v₁ = [1, 5, 0] and v₂ = [-3, 0, 2], pick a third vector v₃ in R³ such that {v₁, v₂, v₃ } constitutes a basis. (You must show why it is a basis.)
• Do the following matrices form a basis for R^2x2 ? Be sure to explain your answer.
• Find the rank, nullity, basis for N(A) and basis for Range A for the following matrix.
• Prove: If A is an n x n matrix and rank(A) < n, then Ax = 0 has a non-trivial solution.
• Find an orthonormal basis for the subspace of R³ spanned by v₁ = [3, -1, 4] and v₂ = [1, 2, -5].
• Let B₁ be the standard basis {1, t, t²} for the vector space R₂[t] . Find the transition matrix from B₁ to the basis
  • B₂ = {1, 2t - 1, t² + t - 3}. Express the polynomial 4t² - t + 2 in terms of B₂ .

Return to top

Sample Problems for Test #3

• Which of the following are linear transformations? Give reasons.
  • T: R^nxn ---> R given by T(A) = det A
  • T: C¹[0,1] ---> C[0,1] given by T[f(x)] = x f'(x)
  • T: R² ---> R² defined by T(x,y) = (x², y²)
• Let T: R² ---> R₂[t] be the linear transformation defined by T(0,1) = 4 - t + 3t² , T(1,1) = -2 + 5t + t². Find T(5,3).
• Let dimV = n. If a is a non-zero scalar, define T: V ---> V by Tv = av. What is ker T, Range T, nullity(T), rank(T)?
• Prove: If T: V ---> W is a linear transformation, then T is 1-1 iff ker T = {0}.
• Let U, V, and W be finite dimensional vector spaces. If U is isomorphic to V, and V is isomorphic to W, then show that U is isomorphic to W.
• Let B be an invertible nxn matrix. Define the linear transformation T: R^nxn ---> R^nxn by T(A) = AB. What are the rank and nullity of T?
• Let T: R³ ---> R₂[t] be defined by T(a,b,c) = at² - (b+c)t + 2b. If C = {2, 3t + 1, t² - t }, find the matrix representation of T with respect to the standard basis in R³ and the basis C in R₂[t].
• Prove: If S and T are both linear transformations from vector spaces V to W such that Sv₁ = Tv₁, Sv₂ = Tv₂, . . ., Sv_n = Tv_n for {v₁, v₂ , . . . , v_n } a basis for V, then Sv = Tv for all vectors v in V.
• True/False:
  • An nxn matrix is diagonalizable iff the sum of its geometeric multiplicities is equal to n.
  • If 0 is an eigenvalue of a matrix A, then A is invertible.
  • Similar matrices have the same eigenvalues.
  • The determinant of any square matrix is equal to the sum of its eigenvalues.
• If alpha is an eigenvalue of A, show that (alpha)² is an eigenvalue of A².
• If A is similar to B, show that An is similar to Bn for any positive integer n.
• Find the eigenvalues, eigenspaces, and Jordan canonical form for the matrix with these rows:
  • 2 0 -3
  • -1 4 2
  • 1 -2 0
• Write the possible Jordan forms for a matrix with characteristic equation (x - 5)³(x + 4)⁴(x - 2) = 0 and dim E₅ = 2, dim E_-4 =1.

Useful Websites

http://www.mapleapps.com/index.asp (The Maple Applications Center)

http://www.maa.org (Mathematical Association of America)

http://archives.math.utk.edu (Math Archives)

http://forum.swarthmore.edu (Math Forum)

Marywood University <|> Undergraduate School <|> Undergraduate Admissions

Comments to Dr. Craig M. Johnson, Chair, Dept. of Mathematics: johnsonc@marywood.edu

Last update: October 7, 2002

Copyright © 2003 by Marywood University. All rights reserved. Marywood University, 2300 Adams Avenue, Scranton, PA 18509 (717) 348-6211