MATH-M 303 Lecture Notes - Lecture 4: Identity Matrix, Gaussian Elimination, Augmented Matrix
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If is a matrix containing column vectors and , product defined to be: matrix times vector linear combination, for form , form lc of columns of using corresponding entries of as coefficients, ex. Vector must have same number of rows as matrix has columns: ex. If , express the linear combination as for some matrix and vector . Equation is plane in , so it cannot be consistent for all vectors in space. Can just row reduce coefficient matrix and not worry about augmenting it with: set of vectors spans if every vector is , ie. is consistent for all. Row-column rule (for finding : entry of is sum of the products of corresponding entries from row of and entries of, ex. Compare with previous example of using lc to find product. Matrix is identity matrix- for any , identity matrix has 0"s everywhere except 1"s on diagonal and satisfies for.
