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Let W be the subspace of R³ spanned by the vectors 1 -2 Find the matrix A of the orthogonal projection onto W. A = = 6-6 and -8 Problem 21. We know from the lectures that an m × n matrix A defines a linear transformation T from Rn Rm in the following way: T(x) = Ax. Now prove the converse: for any linear transformation from T : Rn → Rm, there exists an m ×n matrix A such that T(x) = Ax. Hint: consider the standard basis vectors of Rn. Where do they get mapped to? Remark. This is why normally people abuse notation and use the same letter to denote both the transformation and the matrix.

Let W be the subspace of R³ spanned by the vectors 1 -2 Find the matrix A of the orthogonal projection onto W. A = = 6-6 and -8 Problem 21. We know from the lectures that an m × n matrix A defines a linear transformation T from Rn Rm in the following way: T(x) = Ax. Now prove the converse: for any linear transformation from T : Rn → Rm, there exists an m ×n matrix A such that T(x) = Ax. Hint: consider the standard basis vectors of Rn. Where do they get mapped to? Remark. This is why normally people abuse notation and use the same letter to denote both the transformation and the matrix.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.CM: Cumulative Review
Problem 4CM
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Let W be the subspace of R³ spanned by the vectors 1 -2 Find the matrix A of the orthogonal projection onto W. A = = 6-6 and -8
Transcribed Image Text:Let W be the subspace of R³ spanned by the vectors 1 -2 Find the matrix A of the orthogonal projection onto W. A = = 6-6 and -8
Problem 21. We know from the lectures that an m × n matrix A defines a linear transformation T from Rn Rm in the following way: T(x) = Ax. Now prove the converse: for any linear transformation from T : Rn → Rm, there exists an m ×n matrix A such that T(x) = Ax. Hint: consider the standard basis vectors of Rn. Where do they get mapped to? Remark. This is why normally people abuse notation and use the same letter to denote both the transformation and the matrix.
Transcribed Image Text:Problem 21. We know from the lectures that an m × n matrix A defines a linear transformation T from Rn Rm in the following way: T(x) = Ax. Now prove the converse: for any linear transformation from T : Rn → Rm, there exists an m ×n matrix A such that T(x) = Ax. Hint: consider the standard basis vectors of Rn. Where do they get mapped to? Remark. This is why normally people abuse notation and use the same letter to denote both the transformation and the matrix.
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