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.

The code that generates the quiz can be viewed, but it isn't necessary to do so.

Consider the vectors given:

In[230]:=

Out[230]=

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[231]:=

Out[231]//MatrixForm=

Out[232]//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.

Solutions

Created by Mathematica  (April 20, 2006)