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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 (B)
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 (A)
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 (A)
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.
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.
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.
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?
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.
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