Math 304 Solutions 6

Math 304 Solutions 6 - Math 304 Solutions 6 1 Section 4.1...

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Math 304 Solutions 6 1 Section 4.1 17. (b) Determine the kernel and range of the following linear operator on R 3 . L ( x ) = ( x 1 ,x 2 , 0) T Answer: This linear transformation is the same as multiplication by the matrix 1 0 0 0 1 0 0 0 0 . The kernel of this linear transformation is the same as the nullspace of the matrix. Thus, the kernel of L is spanned by the vector 0 0 1 . The range of this linear transformation is the same as the column space of the matrix. Thus, the range of L is spanned by the vectors 1 0 0 and 0 1 0 . 21. A linear transformation L : V W is said to be one-to-one if L ( v 1 ) = L ( v 2 ) implies that v 1 = v 2 (i.e., no two distinct vectors v 1 , v 2 get mapped into the same vector w W ). Show that L is one-to-one if and only if ker( L ) = { 0 V } . Answer: First, we show that if L is one-to-one, then ker( L ) = { 0 V } : Since L is one-to-one, there can be at most one vector that maps to 0 W . The only vector that maps to
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Math 304 Solutions 6 - Math 304 Solutions 6 1 Section 4.1...

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