A Course in Linear Algebra by David B. Damiano PDF
By David B. Damiano
The authors outline normal vector areas and linear mappings on the outset and base all next advancements on those strategies. This method offers a ready-made context, motivation, and geometric interpretation for every new computational strategy. Proofs and summary problem-solving are brought from the beginning, delivering scholars a right away chance to perform making use of what they have discovered. every one bankruptcy includes an advent, precis, and supplementary routines. The textual content concludes with a couple of important appendixes and options to chose exercises.
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Additional resources for A Course in Linear Algebra
Y G W, we must have x + y E W. If W has this property, we say that the set W is closed under addition. 2. For all x G W and all c G R, we must have ex G W . If W has this property, we say that W is closed under scalar multiplication. 3. The zero vector of V must be contained in W. 6a). In fact, it is possible to condense these three conditions into a single condition that is easily checked and that characterizes the subspaces of V completely. 8) Theorem . Let V be a vector space, and let W be a nonempty subset of V.
Let m, n > 1 be integers. 3. LINEAR COMBINATIONS Here a,j represents the entry in the rth row and the jth column of the matrix. The set of all m b y n matrices with real entries will be denoted by Mmx„(R). We usually use the shorthand notation A = («,,) to indicate the matrix whose entries are the a,j. If A = (a,y)and5 = (by) are both matrices in MmX„(R), we can define their sum, denoted A + B, to be the m by n matrix whose entries are the sums of the corresponding entries from A and B. That is, A + B is the matrix whose entries are («,,• + by).
Then cx + 0 = cx G W, so W is closed under scalar multiplication. To see that these observations imply that W is a vector space, note that the axioms 1. 2, and 5 through 8 in Definition ( 1. 1. 1) are satisfied automatically for vectors in W. since they hold for all vectors in V. Axiom 3 is satisfied, since as we have seen 0 G W. 1,6d) ( —I)x = —x G W as well. Hence W is a vector space. ■ To see how the condition of the theorem may be applied, we consider several examples. 2. 10) Example. 8), W is a subspace of R 3 since if x.
A Course in Linear Algebra by David B. Damiano