Questions and Answers
What are the basic properties of subspaces?
A subspace of R^n is any set H in R^n that has three properties: (a) The zero vector is in H. (b) For each u and v in H, the sum u + v is in H. (c) For each u in H and each scalar c, the vector cu is in H.
A subspace is closed under addition and scalar multiplication.
True
What must every subspace include?
The origin.
What is the column space of a matrix A?
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How is the column space Col(A) related to the span?
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The column space of an m x n matrix can be considered in R^n.
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What is the null space of a matrix A?
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The null space of an m x n matrix is a subspace of R^m.
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How do you test if a vector v is in Nul(A)?
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What is a basis for a subspace H of R^n?
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The pivot columns of a matrix A form a basis for the column space of A.
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What does the rank of a matrix define?
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If a matrix A has n columns, then rank(A) + dim(Nul(A)) equals n.
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What can be said about any linearly independent set of exactly p elements in a p-dimensional subspace H?
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If Nul(A) equals the zero vector, then dim(Nul(A)) equals 0.
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Study Notes
Basic Properties of Subspaces
- A subspace H of R^n must contain the zero vector, be closed under vector addition, and closed under scalar multiplication.
Addition and Scaling Subspace Property
- Subspaces require closure under addition and scalar multiplication.
Origin
- Inclusion of the origin is essential for any subspace; lines must pass through the origin to span a subspace in R^n.
Column Space
- The column space of a matrix A consists of all linear combinations of its columns, defined as Col(A).
Column Space Example Explanation
- For a matrix A = [a1...an] with columns in R^m, the column space can be represented as Span {a1,..., an}. It is a subspace of R^m and only equals R^m when its columns span R^m.
Column Space Subspace
- The column space of an m x n matrix is contained in R^m, aligning with the number of elements in each column.
Null Space
- The null space Nul(A) of a matrix A contains all solutions to the homogeneous equation Ax = 0.
Theorem 12
- Null space Nul(A) of an m x n matrix is a subspace of R^n, representing solutions of the system Ax = 0 for m equations and n unknowns.
Null Space Subspace
- The Null Space of an m x n matrix is confined to R^n.
Null Space Process
- To verify if a vector v is in Nul(A), compute Av; if it returns the zero vector, v is in the null space. This implicit definition contrasts with the explicit formulation of the column space.
Basis of a Subspace
- A basis for a subspace H of R^n is a linearly independent set that spans H.
Theorem 13
- Pivot columns of a matrix A represent a basis for its column space, without alteration to their original form.
Dimension of a Subspace
- The dimension of a subspace is determined by the number of vectors comprising it; the zero space {0} is assigned a dimension of 0.
Rank
- The rank of a matrix A is equivalent to the dimension of its column space, represented as Rank(A) = dim(Col(A)). It also equals the number of pivot columns.
Theorem 14
- For a matrix A with n columns, the relationship rank(A) + dim(Nul(A)) = n holds, linking the rank and nullity of the matrix.
Theorem 15
- In a p-dimensional subspace H of R^n, any linearly independent set of p elements automatically forms a basis for H, as does any set that spans H.
Invertible Matrix Theorem - Continued
- An n x n matrix A is invertible if and only if:
- Its columns form a basis for R^n
- Col(A) equals R^n
- dim(Col(A)) equals n
- rank(A) equals n
- Null space Nul(A) is {0}
- dim(Nul(A)) equals 0
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Description
Test your understanding of subspaces in linear algebra with this quiz. Explore the basic properties, addition, and scaling, essential for mastering the concepts of R^n. Perfect for anyone looking to deepen their knowledge in this fundamental topic.
