Questions and Answers
If B={v1,...,vp} is a basis for the subspace H and if x=c1v1+...+cpvp, then c1,...,cp are the coordinates of x relative to the basis B.
True
Each line in R^n is a one-dimensional subspace of R^n.
False
The dimension of Col A is the number of pivot columns of A.
True
The dimension of Col A and Nul A add up to the number of columns of A.
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If a set of p vectors spans a p-dimensional subspace H of R^n, then these vectors form a basis for H.
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If B is a basis for a subspace H, then each vector in H can be written in only one way as a linear combination of the vectors in B.
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If B={v1,...,vp} is a basis for a subspace H of R^n, then the correspondence x--> [x]B makes H look and act the same as R^p.
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The dimension of Nul A is the number of variables in the equation Ax=0.
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The dimension of the column space of A is rank A.
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If H is a p-dimensional subspace of R^n, then linearly independent set of p vectors in H is a basis for H.
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Study Notes
Basis and Coordinates
- A basis for a subspace H allows any vector x in H to be expressed uniquely as a linear combination of basis vectors, with coefficients known as coordinates.
Subspaces and Dimensions
- A line in R^n is only a one-dimensional subspace if it passes through the origin; otherwise, it does not qualify as a subspace.
- The dimension of the column space (Col A) equals the number of pivot columns in matrix A.
- The dimension of the column space and the null space (Nul A) together equal the total number of columns in matrix A.
Linear Independence and Spanning
- A collection of p vectors that spans a p-dimensional subspace must also be linearly independent, thereby forming a basis.
- When B is a basis for subspace H, every vector in H can be represented uniquely as a linear combination of the basis vectors.
Correspondence with R^p
- The transformation x → [x]B converts the subspace H into a form that behaves equivalently to R^p, maintaining structure and operations.
Null Space Dimensions
- The dimension of the null space (Nul A) reflects the number of free variables in the equation Ax=0, rather than the total number of variables.
Rank and Column Space
- The rank of matrix A corresponds to the dimension of its column space, establishing a direct link between linear algebra concepts and matrix theory.
Linear Independence in Subspaces
- For a p-dimensional subspace H, any set of p linearly independent vectors located in H serves as a basis for that subspace, reinforcing the relationship between dimension and independence.
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Description
Explore the foundational concepts of basis and coordinates in linear algebra. This quiz covers subspaces, dimensions, linear independence, and their implications in R^n. Test your understanding of how these concepts interconnect and apply to vector spaces.
