Basic skills

(adapted from prof. Sam Buss' Math 20F - Study Outline: Basic Skills Revision 2.0) This is a list of the basic "skills" you should master for Math 20F. I have tried to make the list complete, but of course you are also responsible for items that were inadvertantly omitted. You are expected to know definitions and theorems and how to apply the definitions and theorems appropriately. You are responsible for material from the textbook, and the material covered in class.
• Convert a system of linear equations to matrix form, and vice-versa.
• Convert a matrix to an echelon form (REF), and further to the reduced echelon form (RREF).
• Determine pivot columns, pivot positions, and pivots from an REF.
• Answer the existence and uniqueness quetions.
• Solve an REF system by back substitution.
• Perform row operations.
• Perform vector operations in Rn (vector addition, multiplication by scalar)
• Convert a matrix equation to vector equation, and vice-versa.
• Detemine if a given vector is a linear combination of a given set of vectors.
• Perform matrix operations (addition, multiplication, multiplication by scalar, transpose, power, matrix-vector multiplication).
• Determine if a homogeneous system has a nontrivial solution.
• Determine if a matrix is singular.
• Write a solution set in parametric vector form.
• Determine if a set of vectors is linearly independent.
• Find the matrix of a linear mapping Rn --> Rk.
• Determine if a linear mapping is onto.
• Determine if a linear mapping is one-to-one.
End of Midterm I material
• Determine if A is invertible.
• Compute the inverse of a matrix if it exists.
• Work with elementary matrices and know their correspondence to elementary row operations.
• Put a matrix in LU form if it has an LU form.
• Solve a system using LU factorization.
• Calculate a determinant (using cofactors and row operations).
• Know the effect of row and column operations on the determinant.
• Calculate area or volume of a region using determinant.
• Determine if a subset of a vector space is a subspace (three conditions).
• Know how to use vector space properties. (You do not need to memorise the list of axioms for a vector space.)
• Determine if a given set of vectors is a basis for Rn.
• Work with the vector spaces Rkxn , Pk , C[a,b], Ck[a,b].
• Find a basis for a subspace (Given a set of vectors, find a linearly independent subset).
• Determine the dimension of a subspace.
• Find the row space, column space, and null space of a matrix. Determine the dimensions of these spaces. Find bases for these spaces.
• Calculate the rank of a matrix.
• Calculate the coordinates of a vector in a given basis.
• Perform a change of basis.
• Find the matrices that perform changes of bases (change-of-coordinate matrices, they are the inverses of each other).
End of Midterm II material
• Find the eigenvalues of a matrix.
• Find eigenvectors or the eigenspace corresponding to an eigenvector.
• Determine if a matrix is diagonalizable. If so, find a diagonalization.
• Recognize and use inner product notation.
• Compute inner products. Find the length of a vector. Find the angle between two vectors.
• Find the orthogonal projection of a vector onto another vector.
• Know the complementary properties of Row A and Nul A, and of Col A and Nul AT.
• Find the orthogonal complement of a subspace.
• Determine if a set of vectors is orthogonal (or orthonormal). Determine if a matrix is orthogonal.
• Calculate the coordinates of a vector in a given orthogonal basis.
• Find the projection of a vector b onto a subspace given as the span of orthogonal vectors. (Also, onto a subspace given as the span of orthonormal vectors.)
• Use the matrix method to find the projection of b onto a subspace given as a span of orthogonal vectors.
• Solve least squares problems using the normal equations.
• Solve least squares problems with orthogonal matrix.
• Use the Gram-Schmidt method to find an orthonormal basis of a subspace.
• Find a QR factorization of a matrix.
• Find the projection of a vector \$\vc{b}\$ onto a subspace given as the span of arbitrary vectors.
• Solve least squares problems using the QR factorization.
• Orthogonally diagonalize a given symmetric matrix.