Quiz 2 - Solutions - 1 2 5-4 1 4 2-1-4-1 . Determine...

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Math 54 Quiz 2 Mike Hartglass September 17, 2010 1.) (3 points) Let S = { -→ v 1 , -→ v 2 ,..., -→ v n } be a subset of R n . Without using the phrase “linearly independent”, define what it means for the set S to be linearly dependent . The set S is linearly dependent exactly when there exist real constants x 1 ,...,x n not all zero such that x 1 -→ v 1 + x 2 -→ v 2 + ... + x n -→ v n = -→ 0 . Equivalently, if A is the matrix whose columns are the elements of S then A -→ x = -→ 0 has a nontrivial solution. 2.) (3 points) Let T be the linear transformation of R 2 that rotates points counter- clockwise 90 about the origin then reflects these points through the x 2 - axis. A counterclockwise 90 about the origin sends e 1 to e 2 and the reflection about the x 2 -axis fixes e 2 . Therefore, T ( e 1 ) = e 2 . The rotation sends e 2 to - e 1 and reflection about the x 2 -axis sends - e 1 to e 1 so T ( e 2 ) = e 1 . Therefore the desired matrix is ± 0 1 1 0 ² . 1
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3.) (4 points) Let T be the linear transformation whose matrix is given by
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Unformatted text preview: 1 2 5-4 1 4 2-1-4-1 . Determine whether or not T is one-to-one and whether or not T is onto. Explain. This matrix can be row reduced to 1 2 5-4 0 1 4 2 0 0 0 1 . As there is a pivot in every row, we know that if A represents the above matrix, then A- x =- b has a solution for all- b R 3 so it follows that T is onto. T is not one-to-one; however. One way to see this is that as A contains more column vectors then rows, the column vectors have to be linearly dependent. Another way to see this is that if we solve A- x =- 0 then from the above row reduction, the variable x 3 must be free so we can have nontrivial solutions to A- x =- 0 . 2...
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This note was uploaded on 11/19/2010 for the course LECTURE 1 taught by Professor Yildiz during the Fall '10 term at University of California, Berkeley.

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Quiz 2 - Solutions - 1 2 5-4 1 4 2-1-4-1 . Determine...

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