Quiz

Wednesday, April 26, 2006

Topic: Using the reduced row echelon form to give bases of the null space, row space, and column space.

Instructions: open this notebook in Mathematica. Select (Kernel, Evaluation, Evaluate Notebook) from the menus. This will generate a new quiz with solutions. Select (File, Print...) to print out your quiz. Each quiz is different (some are easier than others) so try a dozen to make sure you understand everything that can happen. None of the problems require any significant amount of computation.

Consider the vectors given:

In[50]:=

Out[50]=

The vectors are a subset of . (Question A) What is the value of d?

Use the following Mathematica calculation to help you answer the questions below.

In[51]:=

Out[51]//MatrixForm=

Out[52]//MatrixForm=

(Question B) What is the dimension of the row space of M?

(Question C) Give a basis for the row space of M.

(Question D) What is the dimension of the column space of M?

(Question E) Give a basis for the column space of M.

(Question F) What is the dimension of the null space of M?

(Question G) Give a basis for the null space of M.

(Question H) What is the dimension of the span of the vectors given in Question A?

(Question I) Give a basis for the span of the vectors given in Question A.

(Question J) Let W be a vector space, and U and V be subspaces. Prove or disprove the following two statements:
(1)    U∩V, the set of all vectors in both U and V, is a subspace of W.
(2)    U∪V, the set of all vectors in either U or V, is a subspace of W.

 Created by Mathematica  (April 20, 2006)