Quotient Spaces and Direct Sums. In what follows, we take V as a
... It is an easy matter to show that x1 , x2 , . . . , xk are linearly independent mod W if and only if x1 , x2 , . . . , xk are linearly independent vectors of V /W . Theorem: Let x1 , x2 , . . . , xr be a basis of V /W , and let w1 , w2 , . . . , ws be a basis for W . Then x1 , . . . , xr , w1 , . . ...
... It is an easy matter to show that x1 , x2 , . . . , xk are linearly independent mod W if and only if x1 , x2 , . . . , xk are linearly independent vectors of V /W . Theorem: Let x1 , x2 , . . . , xr be a basis of V /W , and let w1 , w2 , . . . , ws be a basis for W . Then x1 , . . . , xr , w1 , . . ...
The Inverse of a Square Matrix
... algebra of numbers. In particular, if C is any m n matrix, then CI n C, and if D is any n m matrix, then I n D D. In the regular algebra of numbers, every real number a 0 has a unique multiplicative inverse. This means that there is a unique real number, a 1 such that aa 1 a 1 a 1. ...
... algebra of numbers. In particular, if C is any m n matrix, then CI n C, and if D is any n m matrix, then I n D D. In the regular algebra of numbers, every real number a 0 has a unique multiplicative inverse. This means that there is a unique real number, a 1 such that aa 1 a 1 a 1. ...
