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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q26E (page 120)

What is the image of a function ffrom $鈩�$ to $鈩�$ given by
$f (t) = t^{3} + a t^{2} + b t + c$ ,
where a,b,c are arbitrary scalars?

Short Answer

Expert verified

The image is (f) $= \{f (t) = t^{3} + a t^{2} + b t + c : t i n T\}$ ,image is all of $鈩�$ .

Step by step solution

01

Consider the parameters.

The image of a function consists of all the values the function takes in its target space. If f is a function from X to Y, then

$i m a g e (f) = \{f (x) : i n X\} = \{b i n Y : b = f (x), for some x in X\}$

The Kernel of the transformation results in $\{f (t) = t^{3} + a t^{2} + b t + c : t i n T\}$ , as the reflection is its own inverse.

As the inverse of the reflection exists thus, the image will be $鈩�$ . The kernel is of dimension $i m a g e (f) = \{f (t) = t^{3} + a t^{2} + b t = c : t i n T\}$ so, the image will be of $鈩�$ .

02

Final answer.

The image is $i m a g e (f) = \{f (t) = t^{3} + a t^{2} + b t + c : t i n T\}$ , image is all of $鈩�$ .

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Most popular questions from this chapter

(a) Consider a linear transformation $T$ from ${鈩�}^{5}$ to ${鈩�}^{3}$ . What are the possible values of $d i m (k e r T)$ ? Explain.

(b) Consider a linear transformation $T$ from ${鈩�}^{4}$ to ${鈩�}^{7}$ . What are the possible values of $d i m (i m T)$ ? Explain.

A subspace $V$ of ${鈩�}^{n}$ is called a hyperplane if $V$ is defined by a homogeneous linear equation

$c_{1} x_{1} + c_{2} x_{2} + 鈥� + c_{n} x_{n} = 0$ ,

where at least one of the coefficients $c_{i}$ is nonzero. What is a dimension of a hyperplane in ${鈩�}^{n}$ ? Justify your answer carefully. What is a hyperplane in ${鈩�}^{3}$ ? What is it in ${鈩�}^{2}$ ?

Consider a non-zero vector $\overset{鈫�}{蠀}$ in $\overset{}{{鈩�}^{n}}$ . What is the dimension of the space of all vectors in ${鈩�}^{n}$ that are perpendicular to $\overset{鈫�}{蠀}$ ?

In Exercises 1 through 20, find the redundant column vectors of the given matrix A 鈥渂y inspection.鈥� Then find a basis of the image of A and a basis of the kernel of A.

19. $[1 0 5 3 0 0 1 4 2 0 0 0 0 0 1 0 0 0 0 0]$

Consider linearly independent vectors ${\overset{鈫�}{蠀}}_{1}, {\overset{鈫�}{蠀}}_{2}, . . ., {\overset{鈫�}{蠀}}_{p}$ in a subspaceV of ${鈩�}^{n}$ and vectors $w_{1}, w_{2}, . . ., w_{q}$ that span V. Show that there is a basis ofV that consists of all the ${\overset{鈫�}{u}}_{i}$ and some of the $\overset{鈫�}{w_{j}}$ . Hint: Find a basis of the image of the matrix

$A = [\begin{matrix} | & | & | & | \\ {\overset{鈫�}{v}}_{1} & . . . & {\overset{鈫�}{v}}_{1} & {\overset{鈫�}{w}}_{1} & . . . & {\overset{鈫�}{w}}_{q} \\ | & | & | & | \end{matrix}]$

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