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M294F09WS_LinearSpaces

# M294F09WS_LinearSpaces - B Determine whether or not T is an...

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Math 2940 Worksheet Linear Spaces and Orthogonal Projections (Ch. 4,5) 1. Find the orthogonal projection of 9 ~e 1 onto the subspace of R 4 spanned by 2 2 1 0 and - 2 2 0 1 2. Consider vectors ~v 1 , ~v 2 , and ~v 3 in R 4 . We are told that ~v i · ~v j is the entry a ij of matrix A . 3 5 11 5 9 20 11 20 49 1. Find || ~v 2 || . 2. Find proj ~v 2 ( ~v 1 ), expressed as a scalar multiple of ~v 2 . 3. Find proj V ( ~v 1 ), where V = span ( ~v 2 ,~v 3 ). Express your answer as a linear combination of ~v 2 and ~v 3 . 4. Find the angle enclosed by vectors ~v 2 and ~v 3 .

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3. Find the matrix of the given linear transformation
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Unformatted text preview: B . Determine whether or not T is an isomorphism. If not, ﬁnd a basis for the kernel and image of T , and thus determine the rank of T . (a) T ( z ) = iz from C to C . B = { 1 ,i } (b) T ( f ( t )) = f (-t ) from P 2 to P 2 . B = { 1 ,t,t 2 } (c) T ( M ) = M ± 1 2 0 1 ²-± 1 2 0 1 ² M from U 2 × 2 to U 2 × 2 (the set of all 2 × 2 upper triangular matrices with entries in R ). B = ³± 1 0 0 1 ² , ± 0 1 0 0 ² , ± 1-1 ²´...
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