MATH 461 Midterm: MATH461 BOYLE-M SPRING2006 0101 MID SOL 1
15 Feb 2019
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Math 461-01** spring 2006 exam 2. You may assume given matrix equations are well de ned (i. e. the matrix sizes are compatible): (21 points) given the matrix. Nd a basis for each of the following: (a) col(a), the column space of a. (b) nul(a), the null space of a. (c) row(a), the row space of a. Use the usual operations to row-reduce a to a row-echelon form b: = b . (a) one basis for col(a) is the rst two columns of a, because the rst two columns of. Then with the usual operations on functions, v is a vector space. True (c) the range of a linear transformation (that is, the set of outputs of the transforma- tion) is always a vector space. True (d) a nonempty linearly independent subset of a nite dimensional vector space v is contained in a basis for v .
