Questions and Answers
The row space of A is the same as the column space of A^T.
True
If B is an echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis of Row A.
False
The dimensions of the row space and the column space of A are the same, even if A is not square.
True
The sum of the dimensions of the row space and the null space of A equals the number of rows in A.
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On a computer, row operations can change the apparent rank of a matrix.
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If B is any echelon form of A, the pivot columns of B form a basis for the column space of A.
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Row operations preserve the linear dependence relations among the rows of A.
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The dimension of the null space of A is the number of columns of A that are not pivot columns.
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The row space of A^T is the same as the column space of A.
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If A and B are row equivalent, then their row spaces are the same.
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If Ax = λx for some vector x, then λ is an eigenvalue of A.
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A matrix A is not invertible if and only if 0 is an eigenvalue of A.
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A number c is an eigenvalue of A if and only if the equation (A−cI)x=0 has a nontrivial solution.
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Finding an eigenvector of A may be difficult, but checking whether a given vector is in fact an eigenvector is easy.
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To find the eigenvalues of A, reduce A to echelon form.
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If Ax = λx for some scalar λ, then x is an eigenvector of A.
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If v1 and v2 are linearly independent eigenvectors, then they correspond to different eigenvalues.
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A steady-state vector for a stochastic matrix is actually an eigenvector.
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The eigenvalues of a matrix are on its main diagonal.
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An eigenspace of A is a null space of a certain matrix.
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Study Notes
Row and Column Spaces
- The row space of a matrix A is equal to the column space of its transpose A^T, highlighting the relationship between rows and columns.
- If B is in echelon form and has three nonzero rows, the nonzero rows of B, not necessarily the first three rows of A, form a basis for the row space of A.
- The dimensions of the row space and column space of A are equal, a fact supported by the Rank Theorem, regardless of whether A is square.
Dimensions and Null Spaces
- The sum of the dimensions of the row space and the null space corresponds to the number of columns in A, not the number of rows, aligning with the rank theorem.
- The dimension of the null space consists of the number of columns of A that are not pivot columns, which relate to free variables.
Matrix Operations and Effects
- Row operations can affect apparent rank due to computational errors such as rounding.
- Performing row operations does not maintain the linear dependence relations of rows, as certain operations, like row interchanges, can disrupt these relations.
Eigenvalues and Eigenvectors
- If a matrix equation Ax = λx holds true for a vector x, λ is considered an eigenvalue of A, provided x is a non-zero vector.
- A matrix A is not invertible if and only if 0 is one of its eigenvalues, indicating the presence of a non-trivial kernel.
- A number c is an eigenvalue of A if the equation (A−cI)x=0 yields a nontrivial solution, indicating a relationship with the original eigenvalue equation.
- Determining if a given vector is an eigenvector is simple; verify if Ax is a scalar multiple of x.
Eigenspaces and Steady States
- Eigenvalues are not generally present on the main diagonal of a matrix, except in triangular matrices.
- A steady-state vector associated with a stochastic matrix functions as an eigenvector, characterized by the equation Axx = x with λ equal to 1.
- The eigenspace of a matrix A is defined as the null space of the matrix A−λI, connecting eigenvalues to linear transformations.
Row Equivalence and Its Consequences
- When two matrices, A and B, are row equivalent, they exhibit identical row spaces, simplifying the process of finding the row space through echelon forms.
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Description
This quiz covers the fundamental concepts of row and column spaces in linear algebra, including their dimensions and relationships as described by the Rank Theorem. It explores the implications of matrix operations on these spaces and the understanding of null spaces. Perfect for students studying linear algebra.
