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Published bySimon Clarke Modified over 7 years ago
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REVIEW Linear Combinations Given vectors and given scalars
is a linear combination of with weights Example:
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REVIEW A vector equation
has the same solution set as the linear system whose augmented matrix is can be generated by a linear combination of vectors in if and only if the following linear system is consistent:
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REVIEW Definition If , then the set of all linear combinations of is denoted by Span and is called the subset of spanned by
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REVIEW Example:
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1.4 The Matrix Equation
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Key Idea We will see how think of a linear combination of vectors as a product of a matrix and a vector.
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Linear Combinations is a linear combination of the columns of A with corresponding entries in x as weights. Note: is defined only if the number of columns of A equals the number of elements in .
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Example:
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Compute
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A matrix equation has the same solution set as the vector equation which has the same solution set as the linear system whose augmented matrix is Therefore: Ax = b has a solution if and only if b is a linear combination of columns of A
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The following statements are equivalent:
Theorem 4: The following statements are equivalent: For each vector b, the equation has a solution. 2. Each vector b is a linear combination of the columns of A. 3. The columns of A span 4. A has a pivot position in every row. Note: Theorem 4 is about a coefficient matrix A, not an augmented matrix.
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Row-Vector Rule for Computing
If the product is defined, then the ith entry in is the sum of the products of corresponding entries from row i of A and from the vector x.
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Theorem 5:
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