If an m by n matrix A is row equivalent to an m by n matrix B, then the row space of A is equal to the row space of B. Explain why row equivalent matrices have the same row space. The column space of A is the subspace of R.m spanned by the column vectors of A. What is the dimension of R.n with the standard operations? it is just n What is the dimension of P.n with the standard operations? it is just n + 1 What is the dimension of Matrix m.n with the standard operations? it is just m.n Define row space and column space Let A be an m by n matrix.ġ.The row space of A is a subspace of R.n spanned by the row vectors of A.Ģ. When V consists of the zero vector alone, the dimension of V is defined as zero. if a vector space V has a basis consisting of n vectors, then the number n is called the dimension of V, denoted by dim (V) = n. If a vector space V has one basis with n vectors, then every basis for V has n vectors. Explain the number of vectors in a basis. Explain Bases and Linear Dependence If S is a basis for a vector space V, then every set containing more than n vectors in V is linearly dependent. what is the uniqueness of basis representation? If S is a basis for a vector space V, then every vector in V can be written in one and only one way as a linear combination of vectors in S. If the coefficient matrix of the system has a nonzero determinant, what does it say for the solution? It is trivial. Because the coefficient matrix of a system has a nonzero determinant, what can we infer? That the system has a unique solution. Define Basis a set of vectors S, in a vectors space V is called a basis for V when the following conditions are true:Ģ. what is corollary? two vectors u and v in a vector space V are linearly dependent if and only if one is a scalar multiple of the other. what is a one property of linearly dependent sets? a set S with vector units greater or equal to two, is linearly dependent if and only if at least one of the vectors can be written as a linear combination of the other vectors in S. Check the determinants, use Gaussian elimination, and or check for unique solutions. How do you test for linear dependence and independence. A trivial solution equates all coefficients to zero which when multiplied by a vector does not create a linear combination. Explain the trivial solution meaning linearly independent case. They are linearly independent when they have a trivial solution with a nonzero determinant. True or False, a set of vectors are linearly independent when they have a nontrivial solution. If S is a set of vector space V, then the span of S is the set of all linear combinations of the vectors in S. If S is a subset of a vector space, it is called a spanning set of V when every vector in V can be written as a linear combination of vectors in S. When can I vector be called a linear combination of other vectors? When it can be written as equaling a combination of the other vectors corresponding vector components times a scalar. It consists of all points in a plane that pass through the origin.ģ. It consists of all points on a line that passes through the origin.ģ. If a Vector is a subset of R 2 then it is a subspace if and only if one of the following three possibilities is true. W consists of all points on a line that passes through the origin.ģ. What is the theorem for the intersection of two sub spaces? If two vectors are both subspaces then their intersection is also a subspace of their governing vector. When is a square matrix symmetric? When it is equal to its transpose. What is a proper subspace? Any subspaces that are not the zero vector or the vector itself. Name the two trivial subspaces? The zero vector and the vector itself. If u is in W and c is any scalar, then the product of c and u are in W. What is the second test for subspace? If W is a non-empty subset of a vector space V then W is a subspace a V if and only if the following closure conditions hold:Ģ. If u and v are in W then the sum of u plus v is in W. What is the first test for subspace? If W is a non-empty subset of a vector space V then W is a subspace a V if and only if the following closure conditions hold:ġ. A non-empty subset W of a vector space V is called a subspace of V when W is a vector space under the operations of a addition and scalar multiplication defined in V. A vector space consists of what for entities? A set of vectors, a set of scalars, and two operations.
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