1. [Consistent systems of equations?] (/10) Consider the problem of determining whether the following system of equations is consistent for all b1, b2, b3 2x1 鮶 4x2 鮶 2x3 = b1 鯦5x1 + x2 + x3 = b2 7x1 1 5x2 ∈ 3x3 = 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 ma￾trix 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.  