# Section-by-section competencies for MAT 233, Linear Algebra

## Chapter 1

The review page for In-Class Assessment 1 serves as the competency list for this chapter.

## Chapter 2: Matrix Algebra

• 2.1: Matrix Operations
• Identify the (i,j)-entry of a matrix.
• Identify the diagonal entries of a matrix.
• Identify the main diagonal of a matrix.
• Determine whether a matrix is diagonal or not.
• Write the n x n zero matrix and the n x n identity matrix (denoted I_n).
• Create the n x n zero and identity matrices in MATLAB using the ZERO and EYE commands.
• Determine whether two matrices are equal.
• Add two matrices of equal size.
• Calculate a scalar multiple of a matrix.
• Use Theorem 2.1 to calculate sums and scalar multiples of matrices.
• Determine whether two matrices can be multiplied together. If they can, carry out this multiplication by hand and using MATLAB.
• Carry out matrix multiplication using the definition and using the row-column rule.
• State the properties of matrix multiplication as given in Theorem 2.2.
• Raise a matrix to a power.
• Form the transpose of a matrix. (Also be able to do this in MATLAB.)
• EXERCISES FROM LAY: 1--9, 17--20, 39, 40. ALSO -- Mastery Exam MO will assess your skill at these competency areas.
• 2.2: The Inverse of a Matrix
• State the definition of the inverse of a matrix and explain why we cannot think of it as "division".
• Use the term "nonsingular" to refer to an invertible matrix and "singular" to refer to a non-invertible matrix.
• Determine whether a 2x2 matrix is invertible by calculating its determinant. If the matrix is nonsingular, calculate its inverse using Theorem 2.4.
• Solve a system of linear equations using a matrix inverse as in Theorem 2.5.
• State and use the properties of matrix inverses as in Theorem 2.6.
• Give full proofs of the matrix properties in Theorem 2.6, perhaps for a special case (for example, n = 2 or n = 3).
• State the definition of an elementary matrix, and write out an elementary matrix of a given size corresponding to a specific row operation.
• Explain how repeated multiplication of elementary matrices corresponds to row-reduction.
• Find the inverse of an elementary matrix.
• State Theorem 2.7 and use its result to find the inverse of an nxn matrix (or to determine that such a matrix is singular).
• EXERCISES FROM LAY: 1--7, 9, 17, 18, 21--24, 29--33.
• 2.3: Characterizations of Invertible Matrices
• State all 12 parts of the Invertible Matrix Theorem as given in this section.
• Explain the logical equivalency of any two parts of the Invertible Matrix Theorem. (Being able to prove that logical equivalency is an excellent way to explain it.)
• EXERCISES FROM LAY: 1--10 (don't just calculate A^(-1) in MATLAB; use the results of the IVT, for example by determining that the columns of the matrix are linearly dependent or span all of R^3); 11, 13, 15--24.
• 2.5: Matrix Factorizations
• Define what an LU factorization (sometimes called an LU "decomposition") of a matrix is.
• Explain why an LU factorization for a matrix is useful.
• Use the tic and toc commands in MATLAB to find the time elapsed during a MATLAB calculation and use these commands to compare the time needed to solve a linear system using "straight" row reduction versus using an LU factorization.
• Find an LU factorization of a matrix by hand.
• Find an LU factorization of a matrix using the LU command in MATLAB.
• Solve a linear system using an LU factorization.
• EXERCISES FROM LAY: 1, 3, 5, 7--18.
• 2.8: Subspaces of Rn
• State the definition of a subspace of Rn.
• Determine whether a given subset of Rn is or isn't a subspace of Rn.
• State the definitions of the column space and null space of a matrix.
• Determine whether a given vector is in the column space of a matrix.
• Determine whether a given vector is in the null space of a matrix.
• State the definition of a basis for a subspace of Rn.
• Find a basis for the column space of a matrix.
• Find a basis for the null space of a matrix.
• EXERCISES FROM LAY: 3, 5 ("generated" means "spanned"), 7, 9, 11, 13, 15, 17, 19, 21, 23, 25.
• 2.9: Dimension and Rank
• Find the coordinate vector for a given vector relative to a given basis.
• State the definition of the dimension of a subspace of Rn.
• State the definition of the rank of a matrix.
• Find the dimension of a subspace of Rn.
• Find the rank of a given matrix.
• State the Rank Theorem (Theorem 2.14).
• State the Basis Theorem (Theorem 2.15).
• State the six new parts of the Invertible Matrix Theorem.
• EXERCISES FROM LAY: 1--13 odd, 17, 19, 21, 23, 25, 29.

## Chapter 3: Determinants

• 3.1: Introduction to Determinants
• Calculate the determinant of a 2x2 matrix by hand.
• Calculate the determinant of a 3x3, 4x4, etc. matrix by hand, using an expansion along any row or any column. Make intelligent choices of which column/row to use, and get the sign pattern right.
• Find the determinant of an upper- or lower-triangular matrix without resorting to the full definition of the determinant. (See Theorem 3.2)
• EXERCISES FROM LAY: 1, 5, 9, 13, 25, 27, 39.
• 3.2: Properties of Determinants
• Given the determinant of a matrix A, find the determinant of the matrix obtained from A by any of the three elementary row operations. (See Theorem 3.3)
• Given the determinant of a matrix A, state whether or not A is invertible.
• Given the determinant of a matrix A, draw conclusions about the properties of A based on the Invertible Matrix Theorem. (For example, state whether the columns of A are linearly independent, what the rank of A is, etc.)
• Given the determinant of a matrix A, state the determinant of A^T (the transpose of A).
• Given the determinant of matrices A and B, state the determinant of AB.
• Given the determinant of an invertible matrix A, state the determinant of A^{-1}.
• EXERCISES FROM LAY: 5, 9, 15--25 odd, 29 (do this without actually calculating B^5), 35, 39.

Coverage for In-Class Assessment 2 includes all of the above from Chapters 2 and 3.

Chapter 5: Eigenvalues and Eigenvectors
• 5.1: Eigenvectors and Eigenvalues
• State the definitions of eigenvector, eigenvalue, and eigenspace.
• Determine if a given vector is or is not an eigenvector for a matrix.
• Given an eigenvalue for a matrix, find an eigenvector (and a basis for the eigenspace) corresponding to it.
• Find (quickly) the eigenvectors for a triangular matrix.
• State and use the result of Theorem 5.2 about linear independence of eigenvectors corresponding to different eigenvalues.
• EXERCISES FROM LAY: 1--17 odd, 21, 23, 31, 37, 39. (The appropriate MATLAB command for 37 and 39 is eig, but you need to read the documentation on that command to see what it actually does.)
• 5.2: The Characteristic Equation
• Find the characteristic polynomial for a square matrix.
• Use the characteristic polynomial for a matrix to set up the characteristic equation, then solve the characteristic equation to find eigenvalues for a matrix.
• EXERCISES FROM LAY: 1--17 odd (read in the text to learn about "multiplicities"), 21.
• 5.3: Diagonalization
• Raise a diagonal matrix to a power without doing any matrix multiplication.
• State what it means for two matrices to be "similar".
• Raise a matrix to a power without doing more than two matrix multiplications, if it is known that the matrix is similar to a diagonal matrix.
• Determine whether or not a matrix is diagonalizable.
• Diagonalize a matrix if it is determined that the matrix is diagonalizable.
• EXERCISES FROM LAY: 1--21 odd, 33, 35.

## Chapter 6: Orthogonality and Least Squares

• 6.1: Inner Product, Length, and Orthogonality
• Calculate the inner (dot) product of two vectors.
• State and use the properties of the inner product as given in Theorem 6.1.
• Prove one or more of the properties of the inner product as given in Theorem 6.1.
• Calculate the length ("norm") of a vector.

• Determine whether a vector is a unit vector.
• Given a nonzero vector, find a unit vector pointing in the same direction. (I.e., "normalize" the vector.)
• Find the distance between two vectors.
• Determine whether two vectors are orthogonal.
• State the Pythagorean Theorem in terms of vector norms (Theorem 6.2).
• EXERCISES FROM LAY: 1--17 odd, 25, 27.
• 6.2: Orthogonal Sets
• Determine whether a set of vectors forms an orthogonal set.
• Give a partial proof of Theorem 6.4 (orthogonal sets are linearly independent).
• Determine whether a set of vectors is an orthogonal basis for a given subspace.
• Given an orthogonal basis and a vector in the space spanned by that basis, use Theorem 6.5 to find the weights in a linear combination of basis vectors to get the given vector.
• Calculate the projection of a vector onto a line.
• Determine whether a set of vectors is orthonormal (or, whether a set of vectors forms an orthonormal basis for a subspace).
• Determine whether a matrix has orthonormal columns using Theorem 6.6.
• State and use the properties of matrices with orthonormal columns as given in Theorem 6.7.
• EXERCISES FROM LAY: 1--21 odd, 27.
• 6.3: Orthogonal Projections
• Define the concept of the orthogonal complement to a subspace. (This is actually found in section 6.2 of the textbook.)
• Given a subspace W of Rn and a vector y in Rn (NOT necessarily in W), write y as a sum of two vectors -- one of which is in W and the other of which is orthogonal to W. (This uses the formula in the Orthogonal Decomposition Theorem.)
• If a vector y belongs to a subspace W, find (quickly!) the projection of y into W.
• Calculate the distance from a vector y to a subspace W.
• State Theorem 6.9, the Best Approximation Theorem, and explain what it means in plain English.
• Given an orthonormal basis for a subspace W of Rn and vector y in Rn, find proj_L (y) using the simplified formula in Theorem 6.10.
• EXERCISES FROM LAY: 1--17 odd, 21.
• 6.4: The Gram-Schmidt Process
• Be able to replicate and/or explain Example 2 in the textbook.
• Explain the purpose of the Gram-Schmidt process and give a general outline of how it works.
• Use the Gram-Schmidt process to take a given basis for a subspace of Rn and construct an orthogonal basis for the same subspace.
• Find a QR-factorization for a matrix.
• EYE EXERCISES FROM LAY: 1--13 odd.
• 6.5: Least-Squares Problems
• Define the concept of a least-squares solution to a system.
• Explain the concept of how least-squares solutions are calculated. The process involves orthogonal projections into the column space of a matrix.
• Define the "normal equations" for a system Ax = b and use the normal equations to find the least-squares solution to a system (Theorem 6.13).
• Find the least-squares solution to a system using a QR factorization. (Theorem 6.15)
• EXERCISES FROM LAY: TBA
• 6.6: Application to Linear Models