1. [Consistent systems of equations ] (/10) Consider the problem of determining whether the following system of equations is consistent for all b1, b2, b3 2×1 鮶 4×2 鮶 2×3 = b1 鯦5×1 + x2 + x3 = b2 7×1 1 5×2 ∈ 3×3 = b3 1. Define appropriate vectors, and restate the problem in terms of Span { →v1, →v2, →v3}. Then solve that problem. 2. Define an appropriate matrix, and restate the problem using the phrase “columns of A”. an appropriate linear transformation T using the matrix in (2), and restate the problem in terms of T. 1 2. [Solutions on the plane] (/20) 1. Construct a 2×3 matrix A, not in echelon form, such that the solution of A →x = →0 is a line in R3. 2. Construct a 2×3 matrix A, not in echelon form, such that the solution of A →x = →0 is a plane in R3. 3. [Reduced echelon form] (/10) Write the reduced echelon form of a 3 × 3 matrix A such that the first two columns of A are pivot columns and A 鯦321 = 000 4. [Column space and Nullspace] (/20) For all of the below questions, please be as thorough as possible and justify your answer. 1. If P is a 5 × 5 matrix and NulP is the zero subspace, what can you say about solutions of equations of the form P →x = →b for →b in R5 2. If Q is a 4×4 matrix and ColQ = R4 ,what can you say about solutions of equations of the form Q →x = →b for b in R4 3. What can you say about NulB when B is a 54 matrix with linearly independent columns 2 5. [Misc] (/20) For each of the below questions, please be as thorough as possible and justify your answer. 1. Let A be an n×n singular matrix. Describe how to construct an n×n nonzero matrix B such that AB = 0. 2. Given →u ∈ Rn with →u T →u = 1, let P = →u →u T (an outer product) and Q = I I 2P. Justify the following: P2 = P, PT = P, Q2 = I. The transformation →x → P →x is called a projection, and →x → Q →x is called a Householder reflection. Such reflections are used in computer programs to create multiple zeros in a vector (usually a column of a matrix). 6. [Inverse] (/20) Let An be the n × n matrix with 0’s on the main diagonal and 1’s elsewhere. Compute A 1 for n = 4, 5, 6 and make a conjecture about the general form of A 1 for larger values of n.
