MATH 280, Spring 2002 MATH 280, Spring 2002

Theme IV: Vector Spaces and Linear Transformations



The due date for this theme is Friday, March 15, at noon

Reading assignment: Section 4.1, 4.2



A vector space, as defined on page 221, is a collection of objects with an addition and a scalar multiplication that satisfies some standard arithmetic properties. It is not surprising that $\QTR{Bbb}{R}^{n}$ is a vector space; after all, we call the elements of $\QTR{Bbb}{R}^{n}$ vectors. On the other hand, it is surprising that the set of all polynomials is a vector space, mostly because we never think of polynomials (or functions) as vectors.

Let $A$ be an $m\times n$ matrix. Then, $A$ is defined on the vector space $\QTR{Bbb}{R}^{n}$. One of the most important subspaces we study in this course is the null space of $A$. What is a subspace of a vector space? What is the null space of a matrix? Let $A$ be the matrix
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Find a spanning set for the null space of $A$.

Let $A$ be an $m\times n$matrix and let $\QTR{bf}{b}$ be a non-zero vector in $\QTR{Bbb}{R}^{m}$ such that the equation MATH is consistent. Let $H $ be the set of solutions of MATH:
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Determine whether or not $H$ is a subspace of $\QTR{Bbb}{R}^{n}$ and justify your answer.

Suppose that Nul MATH. If MATH is consistent, how many solutions does it have? Explain your answer and give an example to demonstrate this situation. Be sure to explain why your example demonstrates the situation

Another important subspace related to a matrix is the column space, which is a subspace of $\QTR{Bbb}{R}^{m}$. If MATH, then
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Let $A$ be the matrix
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Find a linearly independent set of vectors that spans the column space of $A$.

Suppose that MATH is a matrix transformation determined by an $m\times n$ matrix $A$, so that MATH for all MATH. What does it mean to say that $y $ is an element of the range of $T$? If $y$ is in the range of $T$, explain why $y$ is in the column space of $A$.

The concept of a vector space extends to more general settings than Euclidean spaces, $\QTR{Bbb}{R}^{n}$. In the following problem you will study vector spaces of functions.

Let $F$ be the set of all functions MATH that map the interval $\left[ 0,5\right] $ into $\QTR{Bbb}{R}$. Since the sum of functions is a function and the product of a function and a scalar is a function, it follows that $F$ is a vector space. We are interested in studying some of the subsets of $F$. Let $c$ be a fixed real number and consider the following subsets of $F$:
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Show that $F_{0}$ is a subspace of $F$ while $F_{1}$ is not a subspace. Are there any other values $c$ for which $F_{c}$ is a subspace of $F$. Why or why not?

Suppose we change the definition of $F_{c}$ to be
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that is, we evaluate the functions at $3$ instead of $1$. Does that change any of your previous answers about $F_{c}$? Why?

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