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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 4: Q10E (page 176)

Which of the subsets Vof ${鈩�}^{3 脳 3}$ given in Exercise 6through 11are subspaces of ${鈩�}^{3 脳 3}$ . The $3 脳 3$ matrices Asuch that vector $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ is in the kernel of A.

Short Answer

Expert verified

The $3 脳 3$ matrices A such that vector $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ is in the kernel of A is a subspace of ${鈩�}^{3 脳 3}$ .

Step by step solution

01

Definition of subspace.

A subset Wof a linear space Vis called a subspace of Vif

(a) W contains the neutral element 0of V.

(b) W is closed under addition (if fand gare in Wthen so is f + g)

(c) W is closed under scalar multiplication (if fis in Wand Kis scalar, then kfis in W).

we can summarize parts b and c by saying that W is closed under linear combinations.

02

Definition of kernel.

A vector Vis in the kernel of a matrix A if and only if AV= 0.

03

Verification whether the subset is closed under addition and scalar multiplication.

Consider two matrices,

$A = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$

$B = [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$

Evaluatelocalid="1659360041249" $(A + B) [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ .

$(A + B) [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] = A [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] + B [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] = 0 + 0 = 0$

Thus, it is closed under addition. Multiply the matrix by any scalar,

localid="1659408325960" $(c A) [\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}] c = ([A \begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]) = c . 0 = 0$

Thus, it is closed under scalar multiplication. Hence, is a subspace of ${鈩�}^{3 脳 3}$ .

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Most popular questions from this chapter

Which of the subsets $V$ of ${鈩�}^{3 脳 3}$ given in Exercise through 11are subspaces of ${鈩�}^{3 脳 3}$ . Therole="math" localid="1659358236480" $3 x 3$ matrices whose entries are all greater than or equal to zero.

TRUE OR FALSE?

4. The kernel of a linear transformation is a subspace of the domain.

Find the image, kernel and rank of the transformation T in $T (x_{0}, x_{1}, x_{2}, x_{3}, . . . .) = (x_{0}, x_{2}, x_{4}) from V to V .$

Find the basis of all $2 脳 2$ matrix A such that $[\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}] A = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ , and determine itsdimension.

Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphism, $T (x + i y) = x^{2} + y^{2}$ from $鈩�$ to $鈩�$ .

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