Linear Algebra: Subspaces Quiz

Questions and Answers

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.

True

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.

False Signup and view all the answers

If B is an echelon form of a matrix A, then the pivot columns of B form a basis for Col A.

False Signup and view all the answers

Explain why {a1,..., ap} is a basis for Col A if the columns of a matrix A = [a1...ap] are linearly independent.

Because Col A is the set of all linear combinations of a1,..., ap, and since {a1,..., ap} is linearly independent, it is a basis for Col A. Signup and view all the answers

What can you say about Nul F if F is a 5x5 matrix whose column space is not equal to R^5?

Nul F contains a nonzero vector. Signup and view all the answers

If R is a 6x6 matrix and Nul R is not the zero subspace, what can you say about Col R?

Col R is a subspace of R^6, but Col R != R^6. Signup and view all the answers

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.

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.