Mathematical Methods for Social Scientists Math 196 (Sec 45), Winter 2006 Revision Sheet for Mid-term 2

This mid-term will cover those sections in the text book which we have studied in class upto 5.3 and will concentrate on the material covered since the first mid-term. The following questions are of the style you can expect in the exam.

(1) (a) Define what it means for a matrix to be upper triangular, lower triagular or

diagonal.

(b) If A and B are upper triagular matrices, what can you say about AB and A^{t}?

(c) Write

( \begin{matrix}2 & 3 \\ −2 & 4 \end{matrix} )

as a product of elementary matrices.

(2) (a) Give three conditions that are equivalent to a matrix A being invertible. (b) Give a formula for the inverse of a matrix A.

(c) Calculate the inverse of

( \begin{matrix}2 & 6 & 3 \\ −2 & 8 & 4 \\ 0 & −2 & 0 \end{matrix} )

using this formula.

(d) Compute the determinant of

( \begin{matrix}2 & 3 & 4 & 1 \\ 1 & 3 & 2 & 3 \\ 0 & 0 & 2 & 4 \\ 0 & 0 & 4 & 9 \end{matrix} )

(3) (a) Define a subspace. Show that the set of solutions to a system of homogeneous

equations is a subspace.

(b) Define the row space and column space of a matrix. Explain why the row spaces of two row equivalent matrices are the same.

(c) Define what it means for a set of vectors to be linearly independent.

(d) Show that {(0, 1, 1), (1, 0, 1), (1, 1, 0)} is linearly independent.

(4) (a) Define what a basis is for a vector space. Define the dimension of a vector space

— what fact means this definition is sensible?

(b) Give two conditions that are equivalent to being a basis.

(c) Is the set {(−1, 2, 1), (2, 3, 1), (7, 7, 2)} linearly independent? Justify your an-

swer. Find a basis for the span of this set.

It will be useful to review your homework and make sure you understand it. I will also ask questions from the text book.