practice_final_323

# practice_final_323 - 2 satisﬁes L(1 2 1 T =(1 3 T L(2 1-3...

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MATH 323, PRACTICE FINAL Q1. Find the inverse of A = 1 1 2 0 1 3 0 0 1 . Q2. Solve the 2 × 2 matrix equation XA + B = 2 X , where A = ± 3 1 2 5 ² , B = ± 1 0 1 2 ² . Q3. Let A and B be ﬁxed n × n matrices. Let W = { X R n × n : AX = XB } . Prove that W is a subspace of R n × n . Q4. In R 2 × 2 let A = ± 1 1 0 1 ² . Let W = { X R 2 × 2 : AX = XA } . Find a basis for this subspace of R 2 × 2 . Q5. Let B and C be bases for R 2 , and let ± 5 2 2 1 ² be the transition matrix from C to B . Find C when B = ³± 1 1 ² , ± 1 2 ²´ . Q6. The linear transformation L : P 2 R 2 × 2 is deﬁned by L ( p ( x )) = ± p (0) p (1) p (2) p (3) ² . Find the matrix of L when P 2 and R 2 × 2 have respectively the bases B = { x, x + 1 } , C = { E 22 ,E 12 ,E 11 ,E 21 } . 1

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Q7. Let { u 1 ,...,u k } and { w 1 ,...,w n } be bases for subspaces U and W of a vector space V . Prove that { u 1 ,...,u k ,w 1 ,...,w n } is linearly independent if U W = { 0 } . Q8. Deﬁne L : P 3 R 4 by L ( p ( x )) = ( p (0) ,p (0) ,p (1) ,p (0) + p (1)) T . Find bases for the kernel and range of L . Q9. A linear transformation L : R 3 R
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Unformatted text preview: 2 satisﬁes L (1 , 2 , 1) T = (1 , 3) T , L (2 , 1 ,-3) T = (1 , 2) T . Is this enough information to ﬁnd L (4 , 5 ,-1) T ? If YES, ﬁnd it; if NO, ex-plain why it is impossible. Q10. Find eigenvalues and eigenvectors for the matrix A = ± 3 1 1 3 ² . Q11. With A as in Q10, solve Y = AY, Y (0) = (2 , 6) T . Q12. Find the cube root of A = ±-6-14 7 15 ² . Q13. Find the scalar and vector projections of (1 , 2) T onto (1 , 3) T . Q14. Let { x 1 ,x 2 ,x 3 } be nonzero orthogonal vectors in R 3 . Prove that they are linearly independent. Q15. Let S be the span of (1 , 1 , 1) T and (1 , 2 , 3) T . Find S ⊥ . 2...
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practice_final_323 - 2 satisﬁes L(1 2 1 T =(1 3 T L(2 1-3...

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