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Questions and Answers
A subset H of R^n is a subspace if the zero vector is in H.
A subset H of R^n is a subspace if the zero vector is in H.
False
Given vectors v1,...,vp in R^n, the set of all linear combinations of these vectors is a subspace of R^n.
Given vectors v1,...,vp in R^n, the set of all linear combinations of these vectors is a subspace of R^n.
True
The null space of an m x n matrix is a subspace of R^n.
The null space of an m x n matrix is a subspace of R^n.
True
The column space of a matrix A is the set of solutions of Ax = b.
The column space of a matrix A is the set of solutions of Ax = b.
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If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.
If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.
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Explain why {a1,..., ap} is a basis for Col A if the columns of a matrix A = [a1...ap] are linearly independent.
Explain why {a1,..., ap} is a basis for Col A if the columns of a matrix A = [a1...ap] are linearly independent.
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What can you say about Nul F if F is a 5x5 matrix whose column space is not equal to R^5?
What can you say about Nul F if F is a 5x5 matrix whose column space is not equal to R^5?
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If R is a 6x6 matrix and Nul R is not the zero subspace, what can you say about Col R?
If R is a 6x6 matrix and Nul R is not the zero subspace, what can you say about Col R?
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Study Notes
Subspace Definition and Conditions
- A subset H of R^n is considered a subspace only if it contains the zero vector, and is closed under vector addition and scalar multiplication.
- The sum of any two vectors u and v in H, as well as the product of any vector c and u, must also belong to H.
Linear Combinations and Subspaces
- The set of all linear combinations of vectors v1, ..., vp in R^n forms a subspace of R^n.
- This set satisfies the conditions required for a subspace, such as closure under addition and scalar multiplication.
Null Space of a Matrix
- The null space of an m x n matrix is indeed a subspace of R^n.
- Solutions of the equation Ax = 0, where A is an m x n matrix, represent vectors in R^n that satisfy vector space properties.
Column Space Misconceptions
- The column space of a matrix A consists of all vectors b for which the equation Ax = b has at least one solution, not the solutions themselves.
- Understanding the distinction between the column space and solution sets is crucial.
Echelon Forms and Basis
- If B is an echelon form of a matrix A, its pivot columns do not necessarily form a basis for Col A, as their columns may not reside in the original column space.
- It’s important to differentiate between the column spaces of an echelon matrix and the original matrix.
Basis Determination
- If the columns of a matrix A = [a1...ap] are linearly independent, then the set {a1, ..., ap} serves as a basis for Col A.
- The set spans Col A through all linear combinations and maintains linear independence.
Invertible Matrix Theorem Insights
- For a 5x5 matrix F where Col F is not R^5, we conclude that F is non-invertible, indicating the equation Fx = 0 has nontrivial solutions.
- The null space Nul F includes nonzero vectors implying F's failure to span R^5.
Characterizing Column Space vs Null Space
- If Nul R of a 6x6 matrix R includes nonzero vectors, then R cannot be invertible according to the Invertible Matrix Theorem.
- Thus, Col R must be a proper subspace of R^6, not equal to R^6 itself, indicating that the columns do not span the entire space.
Studying That Suits You
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Description
Test your understanding of subspaces in Linear Algebra with these flashcards. Each question prompts you to determine the validity of statements regarding subspace properties and definitions. Perfect for reinforcing your knowledge in R^n.
