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MAT211_Lecture9[1]

MAT211_Lecture9[1] - Review Subspaces of R f:R->R linear...

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1 Review Subspaces of R n f:R m ->R n , linear transformation, im(f) and ker(f). Linear combination. Linear independence. Basis and unique representation. Consider vectors v 1 , v 2 ,.., v m in R n . The vector v i is redundant if v i is a linear combination of v 1 , v 2 ,.., v i-1 . The vectors v 1 , v 2 ,.., v m are linearly independent if none of them is redundant. Suppose that the vectors v 1 , v 2 ,.., v m span a subspace V. If v 1 , v 2 ,.., v m are linearly independent we say that they form a basis of V. If at least one vector v is redundant then v 1 , v 2 ,.., v m are linearly dependent . Theorem. Consider vectors v 1 , v 2 ,.., v p and w 1 , w 2 ,.., w q in a subspace V of R n . If the vectors v 1 , v 2 ,.., v p are linearly independent and the vectors w 1 , w 2 ,.., w q span V then q p . All basis of a subspace V of R n have the same number of vectors. DeFnition: The number of vectors in

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MAT211_Lecture9[1] - Review Subspaces of R f:R->R linear...

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