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Let V be the vector space of 2 × 2 matrices over the real field R. Find a basis and dimension of the subspace W of V spanned by A = 1.B =K 3 В 3 4 C = 5.

Let V be the vector space of 2 × 2 matrices over the real field R. Find a basis and dimension of the subspace W of V spanned by A = 1.B =K 3 В 3 4 C = 5.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let V be the vector space of 2 × 2 matrices over the real field R. Find a basis and dimension of the subspace W of V spanned by A = 1B =K 5 C 3. D = 41
Transcribed Image Text:Let V be the vector space of 2 × 2 matrices over the real field R. Find a basis and dimension of the subspace W of V spanned by A = 1B =K 5 C 3. D = 41
The four matrices A, B, C, D are linearly dependent since holds Hint A + 2B = C The same conclusion can be reached by calculating the determinant or by reducing the matrix with Gauss algorithm. In this case Span (A,B,C,D) = Span(A,B,D) We treat the matrix as if it were a single vector and construct a matrix composed of the vector coefficients of the matrices A, B, D 1 2 -1 3 2 5 1 -1 3 4 -2 The rank of the 3x4 matrix is equal to 3, therefore the three vectors or the matrices A, B, D are a basis of the subspace V. The dimension of V is equal to the number of elements of the base dimV = 3 Note: The determinant of the 3x3 sub-matrix is equal to 7 number other than zero so the rank is 3. 1 2 -1 2 5 1 = 7 3 -2
Transcribed Image Text:The four matrices A, B, C, D are linearly dependent since holds Hint A + 2B = C The same conclusion can be reached by calculating the determinant or by reducing the matrix with Gauss algorithm. In this case Span (A,B,C,D) = Span(A,B,D) We treat the matrix as if it were a single vector and construct a matrix composed of the vector coefficients of the matrices A, B, D 1 2 -1 3 2 5 1 -1 3 4 -2 The rank of the 3x4 matrix is equal to 3, therefore the three vectors or the matrices A, B, D are a basis of the subspace V. The dimension of V is equal to the number of elements of the base dimV = 3 Note: The determinant of the 3x3 sub-matrix is equal to 7 number other than zero so the rank is 3. 1 2 -1 2 5 1 = 7 3 -2
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