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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: Q3E (page 119)

For each matrix $A$ in exercises 1 through 13, find vectors that span the kernel of $A$ . Use paper and pencil.
3. $A = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

Short Answer

Expert verified

The kernel of $A$ is $k e r (A) = s p a n ({\overset{I}{e}}_{1}, {\overset{I}{e}}_{2})$ .

Step by step solution

01

The kernel of a matrix

The kernel of a matrix $A$ is the solution set of the linear system $A \overset{鈫�}{x} = 0$ .

02

Find kernel of the given matrix

Here, the given matrix is in reduced row-echelon form. Notice that the given matrix is a zero matrix. So, the solution set of $A \overset{鈫�}{x} = 0$ are all vectors $\overset{鈫�}{x}$ lies in. ${鈩�}^{2}$

Thus, $\ker (A) = s p a n ({\overset{鈫�}{e}}_{1}, {\overset{鈫�}{e}}_{2})$ .

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Most popular questions from this chapter

Question: 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.

16. $[\begin{matrix} 1 & - 2 & 0 & - 1 & 0 \\ 0 & 0 & 1 & 5 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}]$

In Exercise 40 through 43, 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; see Exercise 53 through 62 in section 1.2. Recall that a conic is a curve in ${鈩�}^{2}$ that can be described by an equation of the form $f (x, y) = c_{1} + c_{2} x + c_{3} y + c_{4} x^{2} + c_{5} x y + c_{6} y^{2} = 0$ , where at least one of the coefficients is non zero.

42. How many conics can you fit through five distinct points $P_{1} (x_{1}, y_{1}), . ......, P_{5} (x_{5}, y_{5})$ ? Describe all possible scenarios, and give an example in each case.

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 ${鈩�}^{2}$ ?

Find a basis of the image of the matrix $[\begin{matrix} 0 & 1 & 2 & 0 & 0 & 3 \\ 0 & 0 & 0 & 1 & 0 & 4 \\ 0 & 0 & 0 & 0 & 1 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]$ .

Consider three linearly independent vectors ${\overset{鈫�}{V}}_{1}, {\overset{鈫�}{V}}_{2}, {\overset{鈫�}{V}}_{3}$ in ${鈩�}^{4}$ .Find $r r e f [\begin{matrix} | & | & | \\ \overset{鈫�}{v_{1}} & \overset{鈫�}{v_{2}} & \overset{鈫�}{v_{3}} \\ | & | & | \end{matrix}]$

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