Final415practice

# Final415practice - x y = 2 2 x-y = 1 y = 2 5 NAME 6[10...

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NAME: Math 415 — Final practice Total points: 100 . Please show your work and explain all answers. Cal- culators, computers, books and notes are not allowed. Suggestion: even if you cannot complete a problem, write out the part of the solution you know. You can get partial credit for it. 1. [10 points] Solve the following linear system. x + y - 2 z + w = 2 x + 2 w = 1 x + 2 y - z = 1 x + y + z + w = 0 1

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NAME: 2. [10 points] Use the Gram-Schmidt method to orthonormalize the fol- lowing vectors in R 4 w 1 = 1 1 0 0 ; w 2 = 1 0 1 0 ; w 3 = 1 0 1 1 2
NAME: 3. [15 points] Explain why the vectors v 1 , v 2 , and v 3 do not form a basis for R 3 . Then ﬁnd a vector w that is orthogonal to the Span ( v 1 , v 2 ). Explain why v 1 , v 2 , and w are a basis for R 3 and why we used orthogonality to ﬁnd w . v 1 = 1 0 1 ; v 2 = 1 2 2 ; v 3 = 1 - 2 0 3

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NAME: 4. [15 points] Find bases for R ( A ) ,R ( A T ), and N ( A ). Which of these subspaces are orthogonal? Verify their orthogonality. A = 1 1 2 - 1 1 0 2 1 0 1 1 1 2 1 4 0 4
NAME: 5. [15 points] Show that the following linear system does not have an exact solution. Then ﬁnd an approximate solution using the least square method.

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Unformatted text preview: x + y = 2 2 x-y = 1 y = 2 5 NAME: 6. [10 points] Prove that only one of these quadratic forms is positive deﬁnite. q 1 ( x ) = 2 x 2 + 3 y 2 + z 2 + 2 xz + 2 yz q 2 ( x ) = 2 x 2 + 6 y 2 + 2 z 2 + 6 xy + 2 xz + 6 yz 6 NAME: 7. [15 points] The matrix A is the standard matrix representation of a lin-ear operator from R 3 to R 3 . Find a basis in which the matrix representation of the same operator is diagonal. Write down also what the diagonal matrix representation is. Explain your answer . A = 1 0 1 0 4 0 1 0 1 7 NAME: 8. [10 points] Determine whether each of the following is a linear transfor-mation from P (2) to P (1) (Prove your answer. Here p ( x ) is a polynomial in P (2) ) a) L ( p ( x )) = p ( x ) + 2 x b) L ( p ( x )) = p 00 ( x )-p ( x ) c) L ( p ( x )) = 2 p ( x )-1 8...
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Final415practice - x y = 2 2 x-y = 1 y = 2 5 NAME 6[10...

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