MAT1503
Linear Algebra I

MAT1503 Study Guide

1. Study Guide Unit 1: Introduction to Systems of Linear Equations
1.1. Identify and analyse a linear equation
1.2. Find the augmented matrix of a system of linear equations
1.3. Determine whether a given sequence of numbers or a given element is a solution of a linear equation (or system of linear equations)
1.4. Determine algebraically or geometrically whether a system of linear equations in 2 unknowns has no solution, exactly one solution or infinitely many solutions
1.5. State the three elementary row operations
1.6. Find the solution of a linear system of equations using elementary row operations
2. Study Guide Unit 2: Gaussian Elimination
2.1. Identify matrices that are in row-echelon form, reduced row-echelon form, or generalized row-echelon form
2.2. Solve a linear system by using Gaussian elimination (i.e. by reducing the augmented matrix to row-echelon form)
2.3. Solve a linear system by using Gauss-Jordan elimination (i.e. by reducing the augmented matrix to reduced row-echelon form)
2.4. Solve a linear system by reducing the augmented matrix to generalized row-echelon form
2.5. Determine if/when a linear system has no solution, exactly one solution or infinitely many solutions
2.6. Determine if/when a linear system of homogeneous equations has only the trivial solution (i.e. only one solution) or the trivial as well as nontrivial solutions (i.e. infinitely many solutions)
3. Study Guide Unit 3: Matrices and Matrix Operations
3.1. Explain what is meant by a matrix and the entries of a matrix
3.2. Determine the size of a matrix
3.3. Determine if/when two matrices are equal
3.4. Find the sum and difference of two matrices
3.5. Determine if matrix addition and subtraction are defined or undefined
3.6. Multiply a matrix by a scalar
3.7. Multiply two matrices
3.8. Determine if the product of two matrices is defined or undefined
3.9. Multiply two matrices where possible
3.10. Write the product of certain types of matrices as a linear combination of row or column matrices
3.11. Determine the transpose of a matrix
3.12. Find the trace of a matrix
3.13. Determine when two matrices are equal
4. Study Guide Unit 4: Inverses; Rules of Matrix Arithmetic
4.1. Apply the various properties of matrix arithmetic
4.2. State which basic rules of arithmetic for real numbers do not hold for matrices
4.3. Explain what is meant by an invertible (non-singular) matrix
4.4. Find the inverse of a 2 × 2 invertible matrix
4.5. Prove and apply the properties of inverse matrices and laws of matrix exponents
4.6. Apply the properties of the transpose of a matrix
5. Study Guide Unit 5: Elementary Matrices and a Method for Finding the Inverse of A
5.1. Explain what is meant by an elementary matrix
5.2. Find for any invertible matrix A, a sequence E₁, E₂, ⋯, Eₖ of elementary matrices such that Eₖ,..E₂,E₁,A=Iₙ
5.3. Determine the inverse A⁻¹ of any invertible matrix A by using the matrix inversion method (which we call the matrix inverse algorithm)
6. Study Guide Unit 6: Further Results on Systems of Equations and Invertibility
6.1. Solve linear systems by using the inverse of its coefficient matrix
6.2. Solve multiple linear systems with the same coefficient matrix simultaneously
6.3. Determine the consistency of a linear system by elimination
6.4. Apply different equivalent statements for the fact that a matrix is invertible, in various problems
7. Study Guide Unit 7: Diagonal, Triangular and Symmetric Matrices
7.1. Identify a matrix, a diagonal matrix, a triangular matrix and a symmetric matrix
7.2. Write down the inverse of an invertible diagonal matrix by inspection
7.3. Understand the effect of the transpose operation on diagonal and triangular matrices
7.4. Understand the effect of inversion on diagonal and triangular matrices
8. Study Guide Unit 8: Determinants by Cofactor Expansion
8.1. Find the minors and cofactors of a square matrix
8.2. Evaluate the determinant of a matrix by cofactor expansion along any row or column of the matrix
8.3. Determine the inverse of an invertible matrix by using its adjoint
8.4. Determine the inverse of a 2×2 matrix by using the determinant
8.5. Solve systems of linear equations by using Cramer's rule
9. Study Guide Unit 9: Evaluating Determinants by Row Reduction
9.1. State the effect that each elementary row operation has on the value of the determinant of the resulting matrix
9.2. Determine, by inspection, the determinants of elementary matrices
9.3. Evaluate the determinant of a matrix by using elementary row operations to reduce the given matrix to an upper triangular matrix
9.4. Evaluate a determinant by using a combination of row or column operations and cofactor expansion
9.5. Evaluate a determinant in terms of a related determinant
10. Study Guide Unit 10: Properties of the Determinant Function
10.1. State and apply the basic properties of determinants
10.2. Analyze how the determinant behaves with respect to basic arithmetic operations
10.3. Use Cramer's rule to solve linear system of equations
10.4. Evaluate a determinant in terms of a related determinant
11. Study Guide Unit 11: Introduction to Vectors, and Norm of a Vector; Vector Arithmetic
11.1. Find the components of a vector given the initial point and terminal point of the vector
11.2. Find the initial or terminal point of a vector given certain information about the vector
11.3. Perform various arithmetic operations on vectors
11.4. Apply the properties of vector arithmetic in 2-space and 3-space
11.5. Find the norm of a vector in 2-space and 3-space
11.6. Find the distance between two points in 2-space and 3-space
12. Study Guide Unit 12: Dot Product; Projections, Cross Product, Lines and Planes in 3-Space, and Euclidean n-Space
12.1. Determine the dot product of two vectors in 2-space or 3-space
12.2. Use the dot product to find the angle (or cosine of the angle) between two vectors in 2-space or 3-space
12.3. Determine if two vectors in 2-space or 3-space are orthogonal (perpendicular)
12.4. Find the orthogonal projection of u on a (the vector component of u along a) and the vector component of u orthogonal to a
12.5. Find the distance between a point and a line in 2-space
12.6. Determine the cross product of two vectors in 3-space
12.7. Use the cross product to find the area of a parallelogram in 3-space
12.8. Use determinants to find the area of a parallelogram in 2-space and the volume of a parallelepiped in 3-space
12.9. Find the equations of lines and planes in 3-space
12.10. Find the distance between a point and a plane or between two parallel planes in 3-space
12.11. Perform various arithmetic operations on vectors in n-space
12.12. Apply the arithmetic properties of vectors in n-space
12.13. Find the Euclidean norm (Euclidean length) of a vector in n-space
12.14. Find the Euclidean distance between two points in n-space
12.15. Determine if two vectors in n-space are orthogonal
13. Study Guide Unit 13: Complex Numbers
13.1. Find the modulus a complex number
13.2. Find the argument of a complex
13.3. Write a complex number in polar form
13.4. Calculate the product and quotient of two complex numbers using the polar form of complex numbers
13.5. State De Moivre's Theorem
13.6. Use De Moivre's Theorem to calculate the roots and/or powers of a complex number
13.7. Linearise trigonometric functions