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Questions and Answers
The cofactor expansion of det A down a column is the negative of the cofactor expansion along a row.
The cofactor expansion of det A down a column is the negative of the cofactor expansion along a row.
False (B)
The determinant of a triangular matrix is the sum of the entries on the main diagonal.
The determinant of a triangular matrix is the sum of the entries on the main diagonal.
False (B)
If the determinant equals 0 (det A = 0), the set of vectors that make up the matrix are linearly dependent.
If the determinant equals 0 (det A = 0), the set of vectors that make up the matrix are linearly dependent.
True (A)
A vector is any element of a vector space.
A vector is any element of a vector space.
If u is a vector in a vector space V, then (−1)u is the same as the negative of u.
If u is a vector in a vector space V, then (−1)u is the same as the negative of u.
A vector space is also a subspace of itself.
A vector space is also a subspace of itself.
A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) the zero vector of V is in H, (ii) u, v, and u + v are in H, and (iii) c is a scalar and cu is in H.
A subset H of a vector space V is a subspace of V if the following conditions are satisfied: (i) the zero vector of V is in H, (ii) u, v, and u + v are in H, and (iii) c is a scalar and cu is in H.
How do you switch a vector in the GIVEN BASIS into a vector in the STANDARD BASIS?
How do you switch a vector in the GIVEN BASIS into a vector in the STANDARD BASIS?
How do you switch a vector in the STANDARD BASIS into a vector in the GIVEN BASIS?
How do you switch a vector in the STANDARD BASIS into a vector in the GIVEN BASIS?
The set ℝ^2 is a two-dimensional subspace of ℝ^3.
The set ℝ^2 is a two-dimensional subspace of ℝ^3.
The number of variables in the equation Ax = 0 equals the dimension of Nul A.
The number of variables in the equation Ax = 0 equals the dimension of Nul A.
A vector space is infinite-dimensional if it is spanned by an infinite set.
A vector space is infinite-dimensional if it is spanned by an infinite set.
If dim V = n and if S spans V, then S is a basis of V.
If dim V = n and if S spans V, then S is a basis of V.
The only three-dimensional subspace of the set of real numbers ℝ^3 is the set of real numbers ℝ^3 itself.
The only three-dimensional subspace of the set of real numbers ℝ^3 is the set of real numbers ℝ^3 itself.
If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.
If B is any echelon form of A, then the pivot columns of B form a basis for the column space of A.
Row operations preserve the linear dependence relations among the rows of A.
Row operations preserve the linear dependence relations among the rows of A.
The dimension of the null space of A is the number of columns of A that are not pivot columns.
The dimension of the null space of A is the number of columns of A that are not pivot columns.
The row space of A^T is the same as the column space of A.
The row space of A^T is the same as the column space of A.
If A and B are row equivalent, then their row spaces are the same.
If A and B are row equivalent, then their row spaces are the same.
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Study Notes
Determinants
- The cofactor expansion of the determinant is equal across any row or column, contrary to the belief that it is negative when expanded down a column.
- For triangular matrices, the determinant is calculated using the product of the entries along the main diagonal, not the sum.
- A determinant equal to zero indicates that the corresponding set of vectors is linearly dependent and the matrix is not invertible.
Vector Spaces
- A vector is defined as any element of a vector space, confirming basic properties of linear algebra.
- The expression (−1)u accurately represents the negative of vector u in a vector space.
- Every vector space qualifies as a subspace of itself as it meets all subspace conditions.
Subspaces
- A subset H of vector space V must include the zero vector, and must satisfy conditions involving all its elements to be a subspace.
- Changing a vector from a given basis to the standard basis requires creating a matrix augmented with the vector and row-reducing.
- To convert a vector from the standard basis to a given basis, multiply and sum the relevant basis vectors by their corresponding components.
Dimensionality
- The set ℝ^2 cannot be a two-dimensional subspace of ℝ^3, as it is not a subset of ℝ^3.
- The quantity of variables in the equation Ax=0 does not directly equate to the dimension of Nul A; rather, free variables determine the dimensionality.
- A vector space with an infinite set of spans is not necessarily infinite-dimensional; its basis may only include a finite number of elements.
Basis and Spanning
- For a set S to be a basis of vector space V, it must both span V and contain as many elements as the dimension of V.
- The only three-dimensional subspace of ℝ^3 is ℝ^3 itself, supported by the Invertible Matrix Theorem regarding linear independence.
Row and Column Spaces
- Pivot columns derived from an echelon form of a matrix A do not necessarily form a basis for the column space of A.
- Linear dependence relations among rows may change with row operations; they do not preserve the original relationships.
- The dimension of the null space of A corresponds to the number of non-pivot columns in A.
- The row space of the transpose matrix A^T aligns with the column space of A, as the rows of A^T are the columns of A itself.
- Row equivalent matrices A and B share the same row space; rows of B are combinations of those in A and vice versa.
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