practice_first_test_323

practice_first_test_323 - k are linearly independent. Q8....

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MATH 323, PRACTICE TEST 1 Q1. Find all solutions of x 1 + 2 x 2 - 3 x 3 = 2 x 1 + 3 x 2 - 4 x 3 = 3 2 x 1 - x 2 - x 3 = - 1 . Q2. Find the inverse of the matrix 1 2 1 - 1 - 1 2 1 2 2 . Q3. Give the definitions of (1) spanning set, (2) linear independence, (3) basis, (4) dimension of a vector space. (5) Write an example of a 2 × 3 matrix in reduced row echelon form. Q4. Find 2 x - 1 as a linear combination of x + 3 and x + 4. Q5. Is ±² x 1 x 2 ³ : x 1 x 2 = 0 ´ a subspace of R 2 ?
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Q6. Is { x 2 + x + 1 , x 2 - x + 2 , 3 x 2 + x + 4 } a spanning set for P 3 ? Q7. If v 1 ,...,v k are linearly independent vectors in R n and A is an invertible n × n matrix, prove that Av 1 ,...,Av
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Unformatted text preview: k are linearly independent. Q8. If { u 1 ,...,u k } and { w 1 ,...,w n } are bases for subspaces U and W of V , and { u 1 ,...,u k ,w 1 ,...,w n } are linearly independent, prove that U W = { } . Q9. Find a basis for the null space of the matrix A = 1 1 1 1 1 2 3 4 . Q10. Let { v 1 ,...,v k } be linearly independent vectors in V , and let v k +1 be a vector not in span { v 1 ,...,v k } . Prove that { v 1 ,...,v k +1 } are linearly independent....
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practice_first_test_323 - k are linearly independent. Q8....

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