Linear Algebra Demystified. Diagnostic Quiz. Diagnostic Questions. Spend about
30 minutes on these questions. Skip ones you can't answer immediately in ...
MIT Tau Beta Pi Teaching Program Linear Algebra Demystified Diagnostic Quiz Diagnostic Questions Spend about 30 minutes on these questions. Skip ones you can’t answer immediately in favor of those you find easiest. Do not be alarmed if the questions seem unfamiliar or if you don’t get very far. If you find you can answer all of these confidently in 30 minutes, please contact the staff regarding employment as a TA for the course, or in the control engineering or digital signal processing industries. Stating Definitions Define the following terms. Feel free to give a few rough words if you cannot give a precise definition. 1. Vector space, subspace 2. Linear combination of a set of vectors, span of a set of vectors 3. Basis of a vector space, dimension of a vector space 4. Eigenvectors of a matrix, eigenvalues of a matrix Correcting Definitions Determine if each of the following definitions is correct. If not, explain why not, and provide either an example of an object covered in the correct definition which the given one misses, or an object permitted by the given definition which is forbidden under the correct definition. 1. A set of vectors S is a subspace of a vector space V if and only if S ⊂ V . 2. A set of vectors S is a subspace of a vector space V if and only if, whenever u, v are in S, then so is u + v. 3. A basis for a vector space V is a subset S of V such that span(S) = V . Checking Definitions 1. Is C3 a vector space? 2. Is {(3, 0, 0), (4, 1, 1)} a spanning set for R3 ? How about {(3, 0, 0), (4, 1, 1), (11, 2, 2)}? 3. Consider the matrix M over R3 given below. Which of the following properties does it satisfy? 1 0 0 (a) Row-echelon-ness. (b) Reduced row-echelon-ness. (c) Upper triangular ness. (d) Invertibility. (e) Orthogonality.
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0 1 0
1 0 1
True/False Determine whether each of the following statements is true or false. If it is true, provide a brief justification. If it is false, provide a counterexample, and correct the statement if you find an obvious way to do so. 1. Some spanning sets are not bases. 2. All matrices are linear maps between two vector spaces. 3. Any linear map can be completely determined by its values on a basis. 4. All diagonalizable matrices have distinct eigenvalues. 5. All singular matrices have zero determinant. 6. A matrix is singular if and only if 1 is an eigenvalue. 7. All maps with determinant 1 are invertible. 8. The rank of a matrix is greater than or equal to its nullity. 9. The sum of the rank and nullity of a matrix is equal to the dimension of its row-space. Applying Key Theorems For each of the following problems, justify your answer by stating or referring to a general theorem or principle. If it is a question, answer it. If it is a statement, determine whether it is true. (Expect a quick/simple answer to each.) 1. If V is a vector space of dimension 3, and S, T are subspaces of dimension 2, then S and T must have at least one nonzero vector in common. 2. If L is a linear transformation from R2 to R3 , then can L have rank 3? 3. Let A be an n × n matrix, and suppose that we can solve the equation A~x = 0 with ~x nonzero. Can A be invertible? 4. Let A be a real matrix with characteristic polynomial p(t) = t2 (t2 + 1). For each of the following properties, can this property be determined by looking at p(t)? If so, then what is its value? • Size
•Trace
• Eigenvectors
•Determinant
•Rank
•Diagonalizability/not
•Nullity
•Eigenvalues
•Symmetric/not
•Orthogonal/not
5. Group the following properties of the real square matrix A into two classes: (1) those which are equivalent to A being an orthogonal matrix, and (2) those which are consequences of A being orthogonal, but are not sufficient to guarantee orthogonality. • All the complex eigenvalues λ of A satisfy |λ| = 1. • A can be diagonalized employing complex vectors, and all |λ| = 1. • All the rows of A are orthonormal vectors. • All the columns of A are orthonormal vectors. • det(A) = ±1.
•||A~v || = ||~v || for every vector ~v . •A> A = I.
• hA~u, A~v i = h~u, ~v i for all vectors ~u, ~v .
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