Explanation Approach: Let S y m 2 ( ℝ ) = { [ a b b c ] : a , b , c ∈ ℝ } Let S k e w 2 ( ℝ ) = { [ 0 a − a 0 ] : a ∈ ℝ } dim [ M n ( ℝ ) ] = n 2 Calculation: The matrix [ a b b c ] can also be written as, [ a b b c ] = a [ 1 0 0 0 ] + b [ 0 1 1 0 ] + c [ 0 0 0 1 ] So S y m 2 ( ℝ ) can be written as, S y m 2 ( ℝ ) = span { [ 1 0 0 0 ] , [ 0 1 1 0 ] , [ 0 0 0 1 ] } As each matrix is linearly independent, so one of the possible basis for S y m 2 ( ℝ ) is { [ 1 0 0 0 ] , [ 0 1 1 0 ] , [ 0 0 0 1 ] } . As the set consists of three matrices, so the dimension of S y m 2 ( ℝ ) will be given as dim [ S y m 2 ( ℝ ) ] = 3 The matrix [ 0 a − a 0 ] can also be written as, [ 0 a − a 0 ] = a [ 0 1 − 1 0 ] As the matrix is linearly independent, so one of the basis for S k e w 2 ( ℝ ) is { [ 0 1 − 1 0 ] }

Differential Equations and Linear Algebra (4th Edition)

4th Edition

ISBN: 9780321964670

Author: Stephen W. Goode, Scott A. Annin

Publisher: PEARSON

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Chapter 4.6, Problem 43P

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The basis for Sym2(ℝ) and Skew2(ℝ), and show that dim[Sym2(ℝ)]+dim[Skew2(ℝ)]=dim[M2(ℝ)].

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Chapter 4.6, Problem 40P

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Differential Equations and Linear Algebra (4th Edition)

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The basis for S y m 2 ( ℝ ) and S k e w 2 ( ℝ ) , and show that dim [ S y m 2 ( ℝ ) ] + dim [ S k e w 2 ( ℝ ) ] = dim [ M 2 ( ℝ ) ] .

Solution Summary: The author explains that the basis for Sym_2(R) and Skew

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To find:

The basis for Sym2(ℝ) and Skew2(ℝ), and show that dim[Sym2(ℝ)]+dim[Skew2(ℝ)]=dim[M2(ℝ)].

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Chapter 4 Solutions

The basis for S y m 2 ( ℝ ) and S k e w 2 ( ℝ ) , and show that dim [ S y m 2 ( ℝ ) ] + dim [ S k e w 2 ( ℝ ) ] = dim [ M 2 ( ℝ ) ] .

Solution Summary: The author explains that the basis for Sym_2(R) and Skew

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To find: The basis for Sym2(ℝ) and Skew2(ℝ), and show that dim[Sym2(ℝ)]+dim[Skew2(ℝ)]=dim[M2(ℝ)].

Want to see the full answer?

Chapter 4 Solutions

To find:

The basis for Sym2(ℝ) and Skew2(ℝ), and show that dim[Sym2(ℝ)]+dim[Skew2(ℝ)]=dim[M2(ℝ)].