# Linear Algebra by Hefferon J. By Hefferon J.

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To see that definition is reasonable, draw a one-dimensional space and make the usual correspondence with R: pick a point to label 0 and another to label 1. 17 times as far. The basic idea here, combining magnitude with direction, is the key to extending to higher dimensions. Section II. Linear Geometry of n-Space 33 An object comprised of a magnitude and a direction is a vector (we will use the same word as in the previous section because we shall show below how to describe such an object with a column vector).

The prior paragraph notes that we will show here that different-looking parametrizations (using the unmodified free variables) describe genuinely different sets. Here is an informal argument that the reduced echelon form version of a matrix is unique. Consider again the example that started this section of a matrix that reduces to three different echelon form matrices. The first matrix of the three is the natural echelon form version. The second matrix is the same as the first except that a row has been halved.

31 [Math. , Sept. 1952] In air a gold-surfaced sphere weighs 7588 grams. It is known that it may contain one or more of the metals aluminum, copper, silver, or lead. When weighed successively under standard conditions in water, benzene, alcohol, and glycerine its respective weights are 6588, 6688, 6778, and 6328 grams. How much, if any, of the forenamed metals does it contain if the specific gravities of the designated substances are taken to be as follows? 00 General = Particular + Homogeneous The prior subsection has many descriptions of solution sets.