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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: Q31E (page 120)

Give an example of a matrixAsuch thatim(A)is the plane with normal vector $[\begin{matrix} 1 \\ 3 \\ 2 \end{matrix}]$ in ${鈩�}^{3} {鈩�}^{3}$ .

Short Answer

Expert verified

$[\begin{matrix} - 2 & - 3 \\ 0 & 1 \\ 1 & 0 \end{matrix}]$ is a matrix A such that $i m (A)$ is the plane with normal vector $[\begin{matrix} 1 \\ 3 \\ 2 \end{matrix}]$ .

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

$i m a g e (f) = \{f (x : x i n X)\} = \{b i n Y : b = f, f o r s o m e x i n X\}$

Consider the system in terms of an equation, $x + 3 y + 2 z = 0 鈬� x = - 3 y - 2 z$

When, $y = 0 鈬� x = [\begin{matrix} - 2 \\ 0 \\ 1 \end{matrix}]$

When, $z = 0 鈬� x = [\begin{matrix} - 3 \\ 1 \\ 0 \end{matrix}]$

Thus, $[\begin{matrix} - 2 \\ 0 \\ 1 \end{matrix}], [\begin{matrix} - 3 \\ 1 \\ 0 \end{matrix}]$ are the required vectors.

02

Final answer.

$[\begin{matrix} - 2 & - 3 \\ 0 & 1 \\ 1 & 0 \end{matrix}]$ is a matrix A such that im(A) is the plane with normal vector $[\begin{matrix} 1 \\ 3 \\ 2 \end{matrix}]$ .

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Most popular questions from this chapter

Can you find a $3 脳 3$ matrix $A$ such that $i m (A) = k e r (A)$ ? Explain.

Consider a nonzero vector $\overset{鈫�}{v}$ in ${鈩�}^{3}$ . Using a geometric argument, describe the image and the kernel of the linear transformation T from ${鈩�}^{3}$ to ${鈩�}^{3}$ given by

$T (\overset{鈫�}{x}) = \overset{鈫�}{v} 脳 \overset{鈫�}{x}$

Consider the matrices

$A = [\begin{matrix} 1 & 0 & 2 & 0 & 4 & 0 \\ 0 & 1 & 3 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 & 6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] and B = [\begin{matrix} 1 & 0 & 2 & 0 & 4 & 0 \\ 0 & 1 & 3 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]$

Show that the kernels of the matrices A and B are different

Consider an n x p matrix A and a p x m matrix B.

a. What can you say about the relationship between rank(A) and rank(AB)?

b. What can you say about the relationship between rank(B) and rank(AB)?

In Exercise 44 through 61, consider the problem of fitting a conic through $m$ given points $P_{1} (x_{1}, y_{1}), . ......, P_{m} (x_{m}, y_{m})$ in the plane. A conic is a curve in ${鈩�}^{2}$ that can be described by an equation of the form $f (x, y) = c_{1} + c_{2} x + c_{3} y + c_{4} x^{2} + c_{5} x y + c_{6} y^{2} + c_{7} x^{3} + c_{8} x^{2} y + c_{9} x y^{2} + c_{10} y^{3} = 0$ , where at least one of the coefficients $c_{i}$ is non-zero. If $k$ is any nonzero constant, then the equations $f (x, y) = 0$ and $k f (x, y) = 0$ define the same cubic.

45. Show that the cubic through the points $(0, 0), (1, 0), (2, 0), (3, 0), (0, 1), (0, 2) 补苍诲鈥� (0, 3)$ can be described by equations of the form $c_{5} x y + c_{8} x^{2} y + c_{9} x y^{2} = 0$ , where at least one of the coefficients $c_{5}, c_{8}, 补苍诲鈥� c_{9}$ is nonzero. Alternatively, this equation can be written as $x y (c_{5} + c_{8} x + c_{9} y) = 0$ . Describe these cubic geometrically.

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