Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 4: Q11E (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 in reduced row-echelon form.

Short Answer

Expert verified

The $3 脳 3$ matrices in reduced row-echelon form is not a subspace of ${鈩�}^{3 脳 3}$ .

Step by step solution

Definition of subspace.

A subsetof 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)

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

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

Consider a matrix which is in reduced row-echelon form, Let

$A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] .$

Multiply the matrix by any scalar, say 3,

$3 . A = 3 [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

$= [\begin{matrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{matrix}]$

The resultant matrix is not in reduced row-echelon form. Thus, it is not closed under scalar multiplication. Hence, it is not a subspace of ${鈩�}^{3 脳 3}$ .

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91影视

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 in reduced row-echelon form.

Short Answer

Step by step solution

Definition of subspace.

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

One App. One Place for Learning.

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91影视

Which of the subsets Vof鈩�3脳3given in Exercise 6through 11are subspaces of鈩�3脳3. The3脳3matrices in reduced row-echelon form.

Short Answer

Step by step solution

Definition of subspace.

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Geometry

Calculus

Logic and Functions

Theoretical and Mathematical Physics

Decision Maths

Study anywhere. Anytime. Across all devices.

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 in reduced row-echelon form.