07win-m1sols

# 07win-m1sols - Math 51 Winter 2007 Midterm 1 Solutions...

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Math 51, Winter 2007 Midterm 1 - Solutions February 1, 2007 (1) (15 points) Complete each of the following sentences. (a) A collection of vectors -→ v 1 , · · · , -→ v m is defined to be linearly dependent if one of the vectors in the collection is a linear combination of the other vectors in the collection. (You could have also said “if there exists scalars c 1 , · · · , c m —not all zero—so that c 1 -→ v 1 + · · · + c m -→ v m = -→ 0 . ) (b) The dimension of a subspace W is defined to be the number of elements in a basis for W . (c) The book lists 4 properties a matrix must have to be in reduced row echelon form. Two of these properties are (i) the leading entry in every row (i.e., ‘pivot’) is 1 and (ii) if a column contains a pivot, then every other entry in the column is 0. You could have also used (iii) if a row has a pivot, any pivots in rows below are further to the right, or (iv) if every entry in a row is 0, then all rows below also have 0 for every entry.
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