3. Let U be the subspace of Rª defined by U = {(x1, x2, X3, X4) E Rª : ¤1 = -22 and x3 = (a) Find two vectors w and w2 such that U = span({w, w2}). (b) Find a basis for U. Extend the basis of U from part (a) to a basis of R4. Hint: See 2.33 from Axler and the corresponding class notes. Start the procedure by considering the linearly dependent set {w1, W2, ē1, ē2, ē3, E4} and removing one of the ē's at a time. Here the e; are the vectors from the standard basis.
Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
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