Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

Answers without the blur.

Just sign up for free and you're in.

Short Answer

Which of the subsets V of $ℝ^{3 x 3}$ given in Exercise 6 through 11 are subspaces of $ℝ^{3 x 3}$ . The diagonal 3x3 matrices.

The diagonal 3x3 matrices subset V of $ℝ^{3 x 3}$ is a subspace of $ℝ^{3 x 3}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Definition of subspace

A subset W of a linear space V is called a subspace of V if

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

(b) W is closed under addition (if f and g are in W then so is f+g )

(c) W is closed under scalar multiplication (if f is in W and k is scalar, then kf is in W).

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

Step 2: Verification whether the subset is closed under addition and scalar multiplication.

Consider two 3x3 matrix namely,

$A = [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] B = [\begin{matrix} - 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 1 \end{matrix}] |A| = 1 (1 - 0) - 0 + 0 |A| = 1$

Find A+B.

$A + B = [\begin{matrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}] + [\begin{matrix} 4 & 0 & 0 \\ 0 & - 3 & 0 \\ 0 & 0 & - 1 \end{matrix}] A + B = [\begin{matrix} 4 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 2 \end{matrix}]$

Thus, the sum of two diagonal matrices is a diagonal matrix, because all other entries will be zero and only the entries in the diagonal can be added and remain non-zero elements.

Similarly, when multiplying a matrix with a scalar, only the non-zero elements in the diagonals can be multiplied and all other will remain as zeros.

$C = 2 [\begin{matrix} 4 & 0 & 0 \\ 0 & - 3 & 0 \\ 0 & 0 & - 1 \end{matrix}] C = [\begin{matrix} 8 & 0 & 0 \\ 0 & - 6 & 0 \\ 0 & 0 & - 2 \end{matrix}]$

Hence, diagonal matrices are a subspace of $ℝ^{3 x 3}$ .

Find Study Materials for

Create Study Materials

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Which of the subsets V of $ℝ^{3 x 3}$ given in Exercise 6 through 11 are subspaces of $ℝ^{3 x 3}$ . The diagonal 3x3 matrices.

Step by Step Solution

Step 1: Definition of subspace

Step 2: Verification whether the subset is closed under addition and scalar multiplication.

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Select your language

Linear Algebra With Applications

Answers without the blur.

Short Answer

Which of the subsets V of ℝ3x3 given in Exercise 6 through 11 are subspaces of ℝ3x3. The diagonal 3x3 matrices.

Step by Step Solution

Step 1: Definition of subspace

Step 2: Verification whether the subset is closed under addition and scalar multiplication.

Most popular questions for Math Textbooks

Want to see more solutions like these?

Recommended explanations on Math Textbooks

Decision Maths

Calculus

Probability and Statistics

Statistics

Mechanics Maths

Geometry

Pure Maths

Which of the subsets V of $ℝ^{3 x 3}$ given in Exercise 6 through 11 are subspaces of $ℝ^{3 x 3}$ . The diagonal 3x3 matrices.