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exam2 - Math 304 Examination 2 Summer 2006 Linear Algebra...

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Math 304 Examination 2 Linear Algebra Summer 2006 Write your name : Answer Key (2 points). In problems 1–5 , circle the correct answer. (5 points each) 1. On the vector space of polynomials, differentiation is a linear operator. True False Solution. True: the derivative of a sum is the sum of the derivatives, and the derivative of a scalar times a polynomial is the scalar times the derivative of the polynomial. 2. If the linear system A x = b is consistent, then the vector b must be in the space N ( A ) . True False Solution. False. If A is an m × n matrix, then b is in R m and N ( A ) is a subspace of R n , so the statement does not even make sense when m negationslash = n . What is true is that b must be in the range R ( A ), and that space is equal to the space N ( A T ) (not N ( A ) ). 3. The matrix parenleftbigg 0 1 1 0 parenrightbigg is the matrix representation (with respect to the standard basis) of the linear operator that reflects each vector x in R 2 about the x 2 axis and then rotates it 90 in the counterclockwise direction. True False Solution. True: the image of the first basis vector parenleftbigg 1 0 parenrightbigg is parenleftbigg 0 1 parenrightbigg , the first column of the matrix, and the image of the second basis vector parenleftbigg 0 1 parenrightbigg is parenleftbigg 1 0 parenrightbigg , the second column of the matrix.
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